File a126671.txt

Notes on Carlo Wood's Polynomials
N. J. A. Sloane and others
Tue, 13 Feb 2007

Part 1.

Carlo Wood (carlo(AT)alinoe.com) wrote to me today
about a family of polynomials that he had constructed,
and asked if there was a simpler description than the 
one he had.

The following is a simple recursive construction of
a family of polynomials that are equivalent to Wood's version
(which is given at the end of this file).

For each n = 1, 2, 3, ... there is an n X n array A[n] of
coefficients. So the arrays A[1], A[2], A[3], ... form an 
infinite square pyramid of numbers.

I work with positive numbers, since the minus
signs in the original version are fully predictable.
Also it seems best to use generating functions
for the columns of A[n] rather than the rows

Here are the first few arrays of coefficients:

A[1]:
  s=0
i
0   1

A[2]:
  s=0    1
i
0   3    1
1   1    1

A[3]:
  s=0    1    2
i
0  11    6    1
1   7    9    2
2   2    3    1

A[4]:
  s=0    1    2    3
i
0  50   35   10    1
1  46   69   26    3
2  26   45   22    3
3   6   11    6    1

A[5]:
  s=0    1    2    3    4
i
0 274  225   85   15    1
1 326  545  270   55    4
2 274  525  320   75    6
3 126  255  170   45    4
4  24   50   35   10    1

(The first 10 rows are given below.)

Let us define f[n,s](x) := Sum_{i=0..n-1} A[n][i,s] x^(n-1-i)
(the generating function for column k of A[n]).

For example, f[4,2] = 6+22*x+26*x^2+10*x^3.

Then the following recurrence defines these arrays:

f[1,0] = 1

f[n,n-1] = (1+x)^(n-1), n > 1,

f[n,s] = n*x*f[n-1,s] + abs(stirling1(n,s+1))*(1+x)^(n-1),

    for n > 1, s < n-1,

where stirling1 is a Stirling number of the first kind.

Here is Maple code to do this:

with(combinat);

f[1,0]:=1;

for n from 2 to 10 do
f[n,n-1]:=sort(expand((1+x)^(n-1)));
for s from 0 to n-2 do
f[n,s]:=sort(expand( n*x*f[n-1,s]+abs(stirling1(n,s+1))*(1+x)^(n-1)));
od; od;

for n from 1 to 10 do
lprint("n=",n);
for s from 0 to n-1 do
lprint(f[n,s]);
od: od:


which gives

"n=", 1
1

"n=", 2
3*x+1
x+1

"n=", 3
11*x^2+7*x+2
6*x^2+9*x+3
x^2+2*x+1

"n=", 4
50*x^3+46*x^2+26*x+6
35*x^3+69*x^2+45*x+11
10*x^3+26*x^2+22*x+6
x^3+3*x^2+3*x+1

"n=", 5
274*x^4+326*x^3+274*x^2+126*x+24
225*x^4+545*x^3+525*x^2+255*x+50
85*x^4+270*x^3+320*x^2+170*x+35
15*x^4+55*x^3+75*x^2+45*x+10
x^4+4*x^3+6*x^2+4*x+1

etc.

The left hand columns of the arrays A[1], A[2], A[3], ...
(in other words, the first polynomial in each set)
are the source for entries A126671 - A126674 in the OEIS


-----------------------------------------------------------

Part 2.

Comment from Vladeta Jovovic, Feb 13 2007:

Sum_{ n >= k } f[n,s]*y^n/n!  =  (-ln(1-(1+x)*y))^(s+1)/((1-x*y)*(1+x)*(s+1)!)


For example:

                                                                 3
                                                ln(1 - (1 + x) y)
                                           -1/6 ------------------
                                                (1 - x y) (1 + x)

expands to give

(x^2 + 2*x + 1)*y^3 + (10*x^3 + 26*x^2 + 22*x + 6)*y^4 + ...
                ---                                ---
                 6                                  24

which is correct.



----------------------------------------------------------

Part 3.

The original version of these polynomials was as follows:


P_0,1(k) = k

P_0,2(k) = k/2 (-k + 3)
P_1,2(k) = k/2 (k - 1)

P_0,3(k) = k/6 (k^2 - 6k + 11)
P_1,3(k) = k/6 (-2k^2 + 9k - 7)
P_2,3(k) = k/6 (k^2 - 3k + 2)

P_0,4(k) = k/24 (-k^3 + 10k^2 - 35k + 50)
P_1,4(k) = k/24 (3k^3 - 26k^2 + 69k - 46)
P_2,4(k) = k/24 (-3k^3 + 22k^2 - 45k + 26)
P_3,4(k) = k/24 (k^3 - 6k^2 + 11k - 6)

P_0,5(k) = k/120 (k^4 - 15k^3 + 85k^2 - 225k + 274)
P_1,5(k) = k/120 (-4k^4 + 55k^3 - 270k^2 + 545k - 326)
P_2,5(k) = k/120 (6k^4 - 75k^3 + 320k^2 - 525k + 274)
P_3,5(k) = k/120 (-4k^4 + 45k^3 - 170k^2 + 255k - 126)
P_4,5(k) = k/120 (k^4 - 10k^3 + 35k^2 - 50k + 24)

and so on.


---------------------------------------------------

Part 4.

The first 12 arrays A[1] through A[12] (see A126682):


n=1
    s=0
i
0     1

n=2
    s=1    0
i
0     1    3
1     1    1

n=3
    s=2    1    0
i
0     1    6   11
1     2    9    7
2     1    3    2

n=4
    s=3    2    1    0
i
0     1   10   35   50
1     3   26   69   46
2     3   22   45   26
3     1    6   11    6

n=5
    s=4    3    2    1    0
i
0     1   15   85  225  274
1     4   55  270  545  326
2     6   75  320  525  274
3     4   45  170  255  126
4     1   10   35   50   24

n=6
          s=5          4          3          2          1          0
i
0           1         21        175        735       1624       1764
1           5         99        755       2745       4640       2556
2          10        186       1300       4170       5890       2844
3          10        174       1120       3270       4270       1956
4           5         81        485       1335       1670        744
5           1         15         85        225        274        120

n=7
          s=6          5          4          3          2          1          0
i
0           1         28        322       1960       6769      13132      13068
1           6        161       1743       9695      28959      43064      22212
2          15        385       3927      20125      53550      67690      30708
3          20        490       4718      22540      55370      65170      28092
4          15        350       3192      14420      33705      38150      16008
5           6        133       1155       5005      11319      12502       5160
6           1         21        175        735       1624       1764        720

n=8
          s=7          6          5          4          3          2          1          0
i
0           1         36        546       4536      22449      67284     118124     109584
1           7        244       3542      27664     124943     323596     435988     212976
2          21        708       9842      72576     303149     704172     815948     351504
3          35       1140      15190     106344     417235     902580     978740     401136
4          35       1100      14070      94136     352275     729260     762580     304464
5          21        636       7826      50400     182189     366324     374444     147120
6           7        204       2422      15120      53263     104916     105588      41040
7           1         28        322       1960       6769      13132      13068       5040

n=9
          s=8          7          6          5          4          3          2          1          0
i
0           1         45        870       9450      63273     269325     723680    1172700    1026576
1           8        351       6564      68166     428568    1662759    3857356    4800564    2239344
2          28       1197      21660     215586    1281756    4612293    9645020   10411884    4292496
3          56       2331      40836     390726    2214240    7523019   14738164   14945364    5868144
4          70       2835      48120     444150    2418654    7880355   14832020   14534100    5562576
5          56       2205      36300     324450    1710744    5407605    9911860    9506700    3582000
6          28       1071      17124     148806     764652    2363319    4251716    4018644    1498320
7           8        297       4620      39186     197232     599193    1063180     994284     367920
8           1         36        546       4536      22449      67284     118124     109584      40320

n=10
          s=9          8          7          6          5          4          3          2          1          0
i
0           1         55       1320      18150     157773     902055    3416930    8409500   12753576   10628640
1           9        485      11340     150690    1251117    6709605   23140710   49127860   57244824   25659360
2          36       1900      43290     556800    4433688   22513260   72175410  138667400  141075576   55988640
3          84       4340      96390    1202160    9222192   44765700  136019310  245888440  235686024   89163360
4         126       6370     137970    1671900   12413898   58121490  169987230  296080400  274689576  101348640
5         126       6230     131670    1553700   11216898   51042390  145259730  246878800  224415576   81542880
6          84       4060      83790     965040    6802992   30269820   84422310  141023960  126418824   45465120
7          36       1700      34290     386400    2669688   11668020   32044410   52849000   46899576   16742880
8           9        415       8190      90510     614817    2648415    7185960   11735540   10335024    3669120
9           1         45        870       9450      63273     269325     723680    1172700    1026576     362880

n=11
            s=10             9             8             7             6             5             4             3             2             1             0
i
0              1            66          1925         32670        357423       2637558      13339535      45995730     105258076     150917976     120543840
1             10           649         18535        306240       3235320      22782837     107974955     338642810     667942220     735979464     318540960
2             45          2871         80300       1292940      13224585      89363043     401407710    1172357010    2099252320    2030120136     779171040
3            120          7524        206140       3238290      32156520     209690712     902454300    2505352410    4235201960    3867983064    1416252960
4            210         12936        347270       5329170      51523230     325984428    1356891690    3635854530    5935135360    5253599736    1876883040
5            252         15246        401170       6022170      56849496     350703738    1422532650    3717051030    5929567952    5146988616    1811429280
6            210         12474        321860       4733190      43747770     264264462    1050523320    2694640410    4229514520    3622621464    1262164320
7            120          6996        177100       2555190      23183160     137613168     538379820    1361628510    2111768120    1791332136     619627680
8             45          2574         63965        906840       8095395      47355462     182894415     457473060     703001860     591974064     203656320
9             10           561         13695        191070       1681680       9716553      37131875      92055480     140435460     117578736      40279680
10             1            55          1320         18150        157773        902055       3416930       8409500      12753576      10628640       3628800

n=12
            s=11            10             9             8             7             6             5             4             3             2             1             0
i
0              1            78          2717         55770        749463       6926634      44990231     206070150     657206836    1414014888    1931559552    1486442880
1             11           846         28963        581790       7606533      67836978     420128929    1801652490    5221552556    9675404376   10157735808    4261576320
2             55          4170        140327       2760450      35173545     303760710    1806030941    7346657670   19857478300   33491516520   30991352832   11545476480
3            165         12330        407913       7864230      97834275     821075310    4717311819   18418747050   47431811460   75723889560   66305530368   23581307520
4            330         24300        790482      14948340     181899630    1488672900    8313859686   31461291180   78365419440  121024556400  102822664032   35695140480
5            462         33516       1072302      19907580     237395466    1900745748   10371310026   38320419060   93233843472  140878920336  117455117472   40178712960
6            462         33012       1039038      18955860     221927706    1743525036    9334038714   33856307100   80964916032  120478279152   99162711648   33587533440
7            330         23220        719202      12906300     148611870    1148592060    6053404566   21639148740   51074707200   75144149520   61275452832   20608076160
8            165         11430        348513       6158130      69856875     532341810    2769288819    9784028430   22857259260   33337488360   26993422368    9030147840
9             55          3750        112607       1961190      21951105     165245850     850273061    2975347650    6893859940    9985714200    8040856032    2678780160
10            11           738         21835        375210       4149453      30906414     157559545     546956190    1258752836    1813140648    1453525920     482630400
11             1            66          1925         32670        357423       2637558      13339535      45995730     105258076     150917976     120543840      39916800


----------------------------------------------------------------

Part 5.

Further comments from Carlo Wood
Date: Wed, 14 Feb 2007 23:05:24 +0100

The Pink polynomials (though I won't object if you keep calling
them Wood's polynomials), are defined by:

The P_{i,n}(k) are polynomials in the variable k of degree n,
defined for 0 <= i < n, such that

P_{i,n}(k) = 0   if  0 <= k <= i
	     1   if  i <  k <= n

They can therefore be written as

P_{i,n}(k) = k/n! \sum_{s=0}^{n-1} c(s,i,n) k^s

Then the coefficients c(s,i,n) are given by:

Mathematica:

c[s_, i_, n_] := n! (-1)^(s+i) Sum[(-1)^(n-r-s-1) StirlingS1[n-r, s+1] Binomial[n-1-r, i]/(n-r)!, {r, 0, n-1-s}];


LaTeX:

c(s,i,n) = (-1)^{s+i} \sum_{r=0}^{n-1-s} {\frac{n!}{(n-r)!} \left|s(n-r,s+1)\right| \binom{n-1-r}{i}}

where s() are the Stirling numbers of the first kind.


Ascii Art:
                     n-1-s
                     -----,           _     _
               (s+i) \         n!   ||  n-r  ||  / n-1-r \ 
c(s,i,n) = (-1)       )      ------ ||       ||  |       |
                     /       (n-r)! ||_ s+1 _||  \   i   /
                     -----`         
                      r=0

C++ code:

// Returns the "coefficients" of the Pink polynomials
// (an additional factor of k/n! is needed).
mpz_class c(int s, int i, int n)
{
  mpz_class sum(0);
  for (int r = 0; r <= n - 1 - s; ++r)
    sum += product(n - r, n) * abs_stirling1(n - r, s + 1) * binomial(n - 1 - r, i);
  if ((s + i) & 1 == 1)
    sum = -sum;
  return sum;
}

These coefficients can be represented as an infinite square pyramid.
Keeping i or s constant, we can look at a slice of it in the form of a
triangle. Since both s and i are in the range [0,n), this will give
rise to 2m slices when we consider n in the range [1,m].

For the sake of helping people finding the Pink polynomials when they
are trying to find large numbers with Google, I'll include all 24 slices
here for values of n up to 12.


s=0

n/i=              0              1              2              3              4              5              6              7              8              9             10             11
1                 1
2                 3             -1
3                11             -7              2
4                50            -46             26             -6
5               274           -326            274           -126             24
6              1764          -2556           2844          -1956            744           -120
7             13068         -22212          30708         -28092          16008          -5160            720
8            109584        -212976         351504        -401136         304464        -147120          41040          -5040
9           1026576       -2239344        4292496       -5868144        5562576       -3582000        1498320        -367920          40320
10         10628640      -25659360       55988640      -89163360      101348640      -81542880       45465120      -16742880        3669120        -362880
11        120543840     -318540960      779171040    -1416252960     1876883040    -1811429280     1262164320     -619627680      203656320      -40279680        3628800
12       1486442880    -4261576320    11545476480   -23581307520    35695140480   -40178712960    33587533440   -20608076160     9030147840    -2678780160      482630400      -39916800

s=1

n/i=              0              1              2              3              4              5              6              7              8              9             10             11
2                -1              1
3                -6              9             -3
4               -35             69            -45             11
5              -225            545           -525            255            -50
6             -1624           4640          -5890           4270          -1670            274
7            -13132          43064         -67690          65170         -38150          12502          -1764
8           -118124         435988        -815948         978740        -762580         374444        -105588          13068
9          -1172700        4800564      -10411884       14945364      -14534100        9506700       -4018644         994284        -109584
10        -12753576       57244824     -141075576      235686024     -274689576      224415576     -126418824       46899576      -10335024        1026576
11       -150917976      735979464    -2030120136     3867983064    -5253599736     5146988616    -3622621464     1791332136     -591974064      117578736      -10628640
12      -1931559552    10157735808   -30991352832    66305530368  -102822664032   117455117472   -99162711648    61275452832   -26993422368     8040856032    -1453525920      120543840

s=2

n/i=              0              1              2              3              4              5              6              7              8              9             10             11
3                 1             -2              1
4                10            -26             22             -6
5                85           -270            320           -170             35
6               735          -2745           4170          -3270           1335           -225
7              6769         -28959          53550         -55370          33705         -11319           1624
8             67284        -323596         704172        -902580         729260        -366324         104916         -13132
9            723680       -3857356        9645020      -14738164       14832020       -9911860        4251716       -1063180         118124
10          8409500      -49127860      138667400     -245888440      296080400     -246878800      141023960      -52849000       11735540       -1172700
11        105258076     -667942220     2099252320    -4235201960     5935135360    -5929567952     4229514520    -2111768120      703001860     -140435460       12753576
12       1414014888    -9675404376    33491516520   -75723889560   121024556400  -140878920336   120478279152   -75144149520    33337488360    -9985714200     1813140648     -150917976

s=3

n/i=              0              1              2              3              4              5              6              7              8              9             10             11
4                -1              3             -3              1
5               -15             55            -75             45            -10
6              -175            755          -1300           1120           -485             85
7             -1960           9695         -20125          22540         -14420           5005           -735
8            -22449         124943        -303149         417235        -352275         182189         -53263           6769
9           -269325        1662759       -4612293        7523019       -7880355        5407605       -2363319         599193         -67284
10         -3416930       23140710      -72175410      136019310     -169987230      145259730      -84422310       32044410       -7185960         723680
11        -45995730      338642810    -1172357010     2505352410    -3635854530     3717051030    -2694640410     1361628510     -457473060       92055480       -8409500
12       -657206836     5221552556   -19857478300    47431811460   -78365419440    93233843472   -80964916032    51074707200   -22857259260     6893859940    -1258752836      105258076

s=4

n/i=              0              1              2              3              4              5              6              7              8              9             10             11
5                 1             -4              6             -4              1
6                21            -99            186           -174             81            -15
7               322          -1743           3927          -4718           3192          -1155            175
8              4536         -27664          72576        -106344          94136         -50400          15120          -1960
9             63273        -428568        1281756       -2214240        2418654       -1710744         764652        -197232          22449
10           902055       -6709605       22513260      -44765700       58121490      -51042390       30269820      -11668020        2648415        -269325
11         13339535     -107974955      401407710     -902454300     1356891690    -1422532650     1050523320     -538379820      182894415      -37131875        3416930
12        206070150    -1801652490     7346657670   -18418747050    31461291180   -38320419060    33856307100   -21639148740     9784028430    -2975347650      546956190      -45995730

s=5

n/i=              0              1              2              3              4              5              6              7              8              9             10             11
6                -1              5            -10             10             -5              1
7               -28            161           -385            490           -350            133            -21
8              -546           3542          -9842          15190         -14070           7826          -2422            322
9             -9450          68166        -215586         390726        -444150         324450        -148806          39186          -4536
10          -157773        1251117       -4433688        9222192      -12413898       11216898       -6802992        2669688        -614817          63273
11         -2637558       22782837      -89363043      209690712     -325984428      350703738     -264264462      137613168      -47355462        9716553        -902055
12        -44990231      420128929    -1806030941     4717311819    -8313859686    10371310026    -9334038714     6053404566    -2769288819      850273061     -157559545       13339535

s=6

n/i=              0              1              2              3              4              5              6              7              8              9             10             11
7                 1             -6             15            -20             15             -6              1
8                36           -244            708          -1140           1100           -636            204            -28
9               870          -6564          21660         -40836          48120         -36300          17124          -4620            546
10            18150        -150690         556800       -1202160        1671900       -1553700         965040        -386400          90510          -9450
11           357423       -3235320       13224585      -32156520       51523230      -56849496       43747770      -23183160        8095395       -1681680         157773
12          6926634      -67836978      303760710     -821075310     1488672900    -1900745748     1743525036    -1148592060      532341810     -165245850       30906414       -2637558

s=7

n/i=              0              1              2              3              4              5              6              7              8              9             10             11
8                -1              7            -21             35            -35             21             -7              1
9               -45            351          -1197           2331          -2835           2205          -1071            297            -36
10            -1320          11340         -43290          96390        -137970         131670         -83790          34290          -8190            870
11           -32670         306240       -1292940        3238290       -5329170        6022170       -4733190        2555190        -906840         191070         -18150
12          -749463        7606533      -35173545       97834275     -181899630      237395466     -221927706      148611870      -69856875       21951105       -4149453         357423

s=8

n/i=              0              1              2              3              4              5              6              7              8              9             10             11
9                 1             -8             28            -56             70            -56             28             -8              1
10               55           -485           1900          -4340           6370          -6230           4060          -1700            415            -45
11             1925         -18535          80300        -206140         347270        -401170         321860        -177100          63965         -13695           1320
12            55770        -581790        2760450       -7864230       14948340      -19907580       18955860      -12906300        6158130       -1961190         375210         -32670

s=9

n/i=              0              1              2              3              4              5              6              7              8              9             10             11
10               -1              9            -36             84           -126            126            -84             36             -9              1
11              -66            649          -2871           7524         -12936          15246         -12474           6996          -2574            561            -55
12            -2717          28963        -140327         407913        -790482        1072302       -1039038         719202        -348513         112607         -21835           1925

s=10

n/i=              0              1              2              3              4              5              6              7              8              9             10             11
11                1            -10             45           -120            210           -252            210           -120             45            -10              1
12               78           -846           4170         -12330          24300         -33516          33012         -23220          11430          -3750            738            -66

s=11

n/i=              0              1              2              3              4              5              6              7              8              9             10             11
12               -1             11            -55            165           -330            462           -462            330           -165             55            -11              1

i=0

n/s=              0              1              2              3              4              5              6              7              8              9             10             11
1                 1
2                 3             -1
3                11             -6              1
4                50            -35             10             -1
5               274           -225             85            -15              1
6              1764          -1624            735           -175             21             -1
7             13068         -13132           6769          -1960            322            -28              1
8            109584        -118124          67284         -22449           4536           -546             36             -1
9           1026576       -1172700         723680        -269325          63273          -9450            870            -45              1
10         10628640      -12753576        8409500       -3416930         902055        -157773          18150          -1320             55             -1
11        120543840     -150917976      105258076      -45995730       13339535       -2637558         357423         -32670           1925            -66              1
12       1486442880    -1931559552     1414014888     -657206836      206070150      -44990231        6926634        -749463          55770          -2717             78             -1

i=1

n/s=              0              1              2              3              4              5              6              7              8              9             10             11
2                -1              1
3                -7              9             -2
4               -46             69            -26              3
5              -326            545           -270             55             -4
6             -2556           4640          -2745            755            -99              5
7            -22212          43064         -28959           9695          -1743            161             -6
8           -212976         435988        -323596         124943         -27664           3542           -244              7
9          -2239344        4800564       -3857356        1662759        -428568          68166          -6564            351             -8
10        -25659360       57244824      -49127860       23140710       -6709605        1251117        -150690          11340           -485              9
11       -318540960      735979464     -667942220      338642810     -107974955       22782837       -3235320         306240         -18535            649            -10
12      -4261576320    10157735808    -9675404376     5221552556    -1801652490      420128929      -67836978        7606533        -581790          28963           -846             11

i=2

n/s=              0              1              2              3              4              5              6              7              8              9             10             11
3                 2             -3              1
4                26            -45             22             -3
5               274           -525            320            -75              6
6              2844          -5890           4170          -1300            186            -10
7             30708         -67690          53550         -20125           3927           -385             15
8            351504        -815948         704172        -303149          72576          -9842            708            -21
9           4292496      -10411884        9645020       -4612293        1281756        -215586          21660          -1197             28
10         55988640     -141075576      138667400      -72175410       22513260       -4433688         556800         -43290           1900            -36
11        779171040    -2030120136     2099252320    -1172357010      401407710      -89363043       13224585       -1292940          80300          -2871             45
12      11545476480   -30991352832    33491516520   -19857478300     7346657670    -1806030941      303760710      -35173545        2760450        -140327           4170            -55

i=3

n/s=              0              1              2              3              4              5              6              7              8              9             10             11
4                -6             11             -6              1
5              -126            255           -170             45             -4
6             -1956           4270          -3270           1120           -174             10
7            -28092          65170         -55370          22540          -4718            490            -20
8           -401136         978740        -902580         417235        -106344          15190          -1140             35
9          -5868144       14945364      -14738164        7523019       -2214240         390726         -40836           2331            -56
10        -89163360      235686024     -245888440      136019310      -44765700        9222192       -1202160          96390          -4340             84
11      -1416252960     3867983064    -4235201960     2505352410     -902454300      209690712      -32156520        3238290        -206140           7524           -120
12     -23581307520    66305530368   -75723889560    47431811460   -18418747050     4717311819     -821075310       97834275       -7864230         407913         -12330            165

i=4

n/s=              0              1              2              3              4              5              6              7              8              9             10             11
5                24            -50             35            -10              1
6               744          -1670           1335           -485             81             -5
7             16008         -38150          33705         -14420           3192           -350             15
8            304464        -762580         729260        -352275          94136         -14070           1100            -35
9           5562576      -14534100       14832020       -7880355        2418654        -444150          48120          -2835             70
10        101348640     -274689576      296080400     -169987230       58121490      -12413898        1671900        -137970           6370           -126
11       1876883040    -5253599736     5935135360    -3635854530     1356891690     -325984428       51523230       -5329170         347270         -12936            210
12      35695140480  -102822664032   121024556400   -78365419440    31461291180    -8313859686     1488672900     -181899630       14948340        -790482          24300           -330

i=5

n/s=              0              1              2              3              4              5              6              7              8              9             10             11
6              -120            274           -225             85            -15              1
7             -5160          12502         -11319           5005          -1155            133             -6
8           -147120         374444        -366324         182189         -50400           7826           -636             21
9          -3582000        9506700       -9911860        5407605       -1710744         324450         -36300           2205            -56
10        -81542880      224415576     -246878800      145259730      -51042390       11216898       -1553700         131670          -6230            126
11      -1811429280     5146988616    -5929567952     3717051030    -1422532650      350703738      -56849496        6022170        -401170          15246           -252
12     -40178712960   117455117472  -140878920336    93233843472   -38320419060    10371310026    -1900745748      237395466      -19907580        1072302         -33516            462

i=6

n/s=              0              1              2              3              4              5              6              7              8              9             10             11
7               720          -1764           1624           -735            175            -21              1
8             41040        -105588         104916         -53263          15120          -2422            204             -7
9           1498320       -4018644        4251716       -2363319         764652        -148806          17124          -1071             28
10         45465120     -126418824      141023960      -84422310       30269820       -6802992         965040         -83790           4060            -84
11       1262164320    -3622621464     4229514520    -2694640410     1050523320     -264264462       43747770       -4733190         321860         -12474            210
12      33587533440   -99162711648   120478279152   -80964916032    33856307100    -9334038714     1743525036     -221927706       18955860       -1039038          33012           -462

i=7

n/s=              0              1              2              3              4              5              6              7              8              9             10             11
8             -5040          13068         -13132           6769          -1960            322            -28              1
9           -367920         994284       -1063180         599193        -197232          39186          -4620            297             -8
10        -16742880       46899576      -52849000       32044410      -11668020        2669688        -386400          34290          -1700             36
11       -619627680     1791332136    -2111768120     1361628510     -538379820      137613168      -23183160        2555190        -177100           6996           -120
12     -20608076160    61275452832   -75144149520    51074707200   -21639148740     6053404566    -1148592060      148611870      -12906300         719202         -23220            330

i=8

n/s=              0              1              2              3              4              5              6              7              8              9             10             11
9             40320        -109584         118124         -67284          22449          -4536            546            -36              1
10          3669120      -10335024       11735540       -7185960        2648415        -614817          90510          -8190            415             -9
11        203656320     -591974064      703001860     -457473060      182894415      -47355462        8095395        -906840          63965          -2574             45
12       9030147840   -26993422368    33337488360   -22857259260     9784028430    -2769288819      532341810      -69856875        6158130        -348513          11430           -165

i=9

n/s=              0              1              2              3              4              5              6              7              8              9             10             11
10          -362880        1026576       -1172700         723680        -269325          63273          -9450            870            -45              1
11        -40279680      117578736     -140435460       92055480      -37131875        9716553       -1681680         191070         -13695            561            -10
12      -2678780160     8040856032    -9985714200     6893859940    -2975347650      850273061     -165245850       21951105       -1961190         112607          -3750             55

i=10

n/s=              0              1              2              3              4              5              6              7              8              9             10             11
11          3628800      -10628640       12753576       -8409500        3416930        -902055         157773         -18150           1320            -55              1
12        482630400    -1453525920     1813140648    -1258752836      546956190     -157559545       30906414       -4149453         375210         -21835            738            -11

i=11

n/s=              0              1              2              3              4              5              6              7              8              9             10             11
12        -39916800      120543840     -150917976      105258076      -45995730       13339535       -2637558         357423         -32670           1925            -66              1


------------------------------------------------------

Part 6:

Postscript from Carlo Wood, Feb 17 2007:

Here is a simpler version of the recursion:
Let us define f[n,s](x) := Sum_{i=0..n-1} A[n][i,s] x^i

Then the following recurrence defines these arrays:

f[n,n-1] = (1+x)^(n-1), n > 1,
f[n,s] = n*f[n-1,s] + abs(stirling1(n,s+1))*(1+x)^(n-1),

Thus x^(n-1-i) ==> x^i, f[1,0]=1 is not needed, and the
factor of *x can be removed.

The Maple program then becomes:

with(combinat);

for n from 1 to 10 do
f[n,n-1]:=sort(expand((1+x)^(n-1)));
for s from 0 to n-2 do
f[n,s]:=sort(expand( n*f[n-1,s]+abs(stirling1(n,s+1))*(1+x)^(n-1)));
od; od;

for n from 1 to 10 do
lprint("n=",n);
for s from 0 to n-1 do
lprint(f[n,s]);
od: od:

which gives

"n=", 1
1

"n=", 2
x+3
x+1

"n=", 3
2*x^2+7*x+11
3*x^2+9*x+6
x^2+2*x+1

"n=", 4
6*x^3+26*x^2+46*x+50
11*x^3+45*x^2+69*x+35
6*x^3+22*x^2+26*x+10
x^3+3*x^2+3*x+1

"n=", 5
24*x^4+126*x^3+274*x^2+326*x+274
50*x^4+255*x^3+525*x^2+545*x+225
35*x^4+170*x^3+320*x^2+270*x+85
10*x^4+45*x^3+75*x^2+55*x+15
x^4+4*x^3+6*x^2+4*x+1

etc.

If we want to take the minus signs into account, the f[] becomes:

g[n,n-1] = (x-1)^(n-1)
g[n,s] = n * g[n-1,s] + StirlingS1[n,s+1](x-1)^(n-1)

then g[n,s] = \sum_{i=0}^{n-1} c(s,i,n) x^i
where c(s,i,n) as defined.