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A123125 Triangle of Eulerian numbers T(n,k), 0<=k<=n, read by rows. 44
1, 0, 1, 0, 1, 1, 0, 1, 4, 1, 0, 1, 11, 11, 1, 0, 1, 26, 66, 26, 1, 0, 1, 57, 302, 302, 57, 1, 0, 1, 120, 1191, 2416, 1191, 120, 1, 0, 1, 247, 4293, 15619, 15619, 4293, 247, 1, 0, 1, 502, 14608, 88234, 156190, 88234, 14608, 502, 1, 0, 1, 1013, 47840, 455192, 1310354 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,9

COMMENTS

The beginning of this sequence does not quite agree with the usual version, which is A173018. - N. J. A. Sloane, Nov 21 2010

Each row of A123125 is the reverse of the corresponding row in A173018. - Michael Somos, Mar 17 2011

A008292 (subtriangle for k>=1 and n>=1 is the main entry for these numbers.

Triangle T(n,k), 0<=k<=n, read by rows given by [0,1,0,2,0,3,0,4,0,5,0,...] DELTA [1,0,2,0,3,0,4,0,5,0,6,...] where DELTA is the operator defined in A084938.

Row sums are the factorials. - Roger L. Bagula and Gary W. Adamson, Aug 14 2008

If the initial zero column is deleted, the result is A008292. - Roger L. Bagula and Gary W. Adamson, Aug 14 2008

This result gives an alternative method of calculating the Eulerian numbers by an Umbral Calculus expansion from Comtet. - Roger L. Bagula, Nov 21 2009

This function seems to be equivalent to the PolyLog expansion. - Roger L. Bagula, Nov 21 2009

A raising operator formed from the e.g.f. of this entry is the generator of a sequence of polynomials p(n,x;t) defined in A046802 that specialize to those for A119879 as p(n,x;-1), A007318 as p(n,x;0), A073107 as p(n,x;1), and A046802 as p(n,0;t). See Copeland link for more associations. - Tom Copeland, Oct 20 2015

The Eulerian numbers in this setup count the number of permutation trees of power n and width k (see the Luschny link). For the associated combinatorial statistic over permutations see the Sage program below and the example section. - Peter Luschny, Dec 09 2015

REFERENCES

L. Comtet, Advanced Combinatorics, Reidel, Holland, 1978, page 245. [Roger L. Bagula, Nov 21 2009]

Douglas C. Montgomery and Lynwood A. Johnson, Forecasting and Time Series Analysis, MaGraw-Hill, New York, 1976, page 91. - Roger L. Bagula and Gary W. Adamson, Aug 14 2008

LINKS

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

Paul Barry, Eulerian polynomials as moments, via exponential Riordan arrays, arXiv preprint arXiv:1105.3043 [math.CO], 2011

P. Barry, Generalized Stirling Numbers, Exponential Riordan Arrays, and Toda Chain Equations, Journal of Integer Sequences, 17 (2014), #14.2.3.

V. Batyrev and M. Blume, The functor of toric varieties associated with Weyl chambers and Losev-Manin moduli spaces, pg. 11, arXiv:/0911.3607 [math.AG], 2009 (From Tom Copeland, Oct 16 2015)

Anna Borowiec, Wojciech Mlotkowski, New Eulerian numbers of type D, arXiv:1509.03758 [math.CO], 2015.

A. Cohen, Eulerian polynomials of spherical type, Münster J. of Math. 1 (2008) (From Tom Copeland, Oct 16 2015)

Tom Copeland, The Elliptic Lie Triad: Ricatti and KdV Equations, Infinigens, and Elliptic Genera

FindStat - Combinatorial Statistic Finder, The number of descents of a permutation.

F. Hirzebruch, Eulerian polynomials, Münster J. of Math. 1 (2008), pp. 9-12.  (From Tom Copeland, Oct 16 2015)

P. Hitczenko and S. Janson, Weighted random staircase tableaux, arXiv preprint arXiv:1212.5498 [math.CO], 2012.

Svante Janson, Euler-Frobenius numbers and rounding, arXiv preprint arXiv:1305.3512 [math.PR], 2013.

A. Losev and Y. Manin, New moduli spaces of pointed curves and pencils of flat connections, arXiv preprint arXiv:math/0001003 [math.AG], 2000 (p. 8). (From Tom Copeland, Oct 16 2015)

Peter Luschny, Permutation Trees

FORMULA

Sum{k,0<=k<=n} T(n,k) = n! = A000142(n).

Sum{k,0<=k<=n} 2^k*T(n,k) = A000629(n).

Sum{k,0<=k<=n} 3^k*T(n,k) = abs(A009362(n+1)).

Sum{k,0<=k<=n} 2^(n-k)*T(n,k) = A000670(n).

Sum_{k, 0<=k<=n}T(n,k)*3^(n-k)=A122704(n). - Philippe Deléham, Nov 07 2007

G.f.: f(x,n)=(1 - x)^(n + 1)*Sum_{k>=0} k^n*x^k. - Roger L. Bagula and Gary W. Adamson, Aug 14 2008

Sum_{k=0..n} T(n,k)*x^k = A000007(n), A000142(n), A000629(n), A123227(n), A201355(n), A201368(n) for x = 0, 1, 2, 3, 4, 5 respectively. - Philippe Deléham, Dec 01 2011

E.g.f. (1-t)/(1-t*exp((1-t)x)). A123125 * A007318 = A130850 = unsigned A075263, related to reversed A028246. A007318 * A123125 = A046802. Evaluating the row polynomials at -1, giving the alternating-sign row sum, generates A009006. - Tom Copeland, Oct 14 2015

EXAMPLE

The triangle T(n, k) begins:

n\k 0 1    2     3      4       5       6      7     8    9 10...

0:  1

1:  0 1

2:  0 1    1

3:  0 1    4     1

4:  0 1   11    11      1

5:  0 1   26    66     26       1

6:  0 1   57   302    302      57       1

7:  0 1  120  1191   2416    1191     120      1

8:  0 1  247  4293  15619   15619    4293    247     1

9:  0 1  502 14608  88234  156190   88234  14608   502    1

10: 0 1 1013 47840 455192 1310354 1310354 455192 47840 1013  1

...  Reformatted. - Wolfdieter Lang, Feb 14 2015

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

The width statistic over permutations, n=4.

[1, 2, 3, 4] => 3; [1, 2, 4, 3] => 2; [1, 3, 2, 4] => 2; [1, 3, 4, 2] => 2;

[1, 4, 2, 3] => 2; [1, 4, 3, 2] => 1; [2, 1, 3, 4] => 3; [2, 1, 4, 3] => 2;

[2, 3, 1, 4] => 2; [2, 3, 4, 1] => 3; [2, 4, 1, 3] => 2; [2, 4, 3, 1] => 2;

[3, 1, 2, 4] => 3; [3, 1, 4, 2] => 3; [3, 2, 1, 4] => 2; [3, 2, 4, 1] => 3;

[3, 4, 1, 2] => 3; [3, 4, 2, 1] => 2; [4, 1, 2, 3] => 4; [4, 1, 3, 2] => 3;

[4, 2, 1, 3] => 3; [4, 2, 3, 1] => 3; [4, 3, 1, 2] => 3; [4, 3, 2, 1] => 2;

Gives row(4) = [0, 1, 11, 11, 1]. - Peter Luschny, Dec 09 2015

MATHEMATICA

f[x_, n_] := f[x, n] = (1 - x)^(n + 1)*Sum[k^n*x^k, {k, 0, Infinity}]; Table[FullSimplify[ExpandAll[f[x, n]]], {n, 0, 10}]; a = Table[CoefficientList[FullSimplify[ExpandAll[f[x, n]]], x], {n, 0, 10}]; Flatten[a] (* Roger L. Bagula, Aug 14 2008 *)

Clear[p, g, m, a];

p[t_] = (1 - x)/(1 - x*Exp[t*(1 - x)]);

a = Table[ CoefficientList[ FullSimplify[ ExpandAll[ n!*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]]], x], {n, 0, 10}];

Flatten[a] (* Roger L. Bagula, Nov 21 2009 *)

PROG

(Haskell)

a123125 n k = a123125_tabl !! n !! k

a123125_row n = a123125_tabl !! n

a123125_tabl = [1] : zipWith (:) [0, 0 ..] a008292_tabl

-- Reinhard Zumkeller, Nov 06 2013

(Sage)

def statistic_eulerian(pi):

    if pi == []: return 0

    h, i, branch, next = 0, len(pi), [0], pi[0]

    while true:

        while next < branch[len(branch)-1]:

            del(branch[len(branch)-1])

        current = 0

        h += 1

        while next > current:

            i -= 1

            if i == 0: return h

            branch.append(next)

            current, next = next, pi[i]

def A123125_row(n):

    L = [0]*(n+1)

    for p in Permutations(n):

        L[statistic_eulerian(p)] += 1

    return L

[A123125_row(n) for n in range(7)] # Peter Luschny, Dec 09 2015

CROSSREFS

See A008292 (subtriangle for k>=1 and n>=1), which is the main entry for these numbers. Another version has the zeros at the ends of the rows, as in Concrete Mathematics: see A173018.

Cf. A007318, A130850, A028246, A046802, A009006.

Sequence in context: A273895 A086329 A085852 * A173018 A055105 A200545

Adjacent sequences:  A123122 A123123 A123124 * A123126 A123127 A123128

KEYWORD

nonn,easy,tabl

AUTHOR

Philippe Deléham, Sep 30 2006

EXTENSIONS

More terms from Roger L. Bagula and Gary W. Adamson, Aug 14 2008

STATUS

approved

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Last modified December 4 15:24 EST 2016. Contains 278750 sequences.