A136212 Triple factorial array, read by antidiagonals, where row n+1 is generated from row n by first removing terms in row n at positions {[m*(m+5)/6], m >= 0} and then taking partial sums, starting with all 1's in row 0.

1, 1, 1, 4, 2, 1, 28, 10, 3, 1, 280, 80, 18, 4, 1, 3640, 880, 162, 28, 5, 1, 58240, 12320, 1944, 280, 39, 6, 1, 1106560, 209440, 29160, 3640, 418, 52, 7, 1, 24344320, 4188800, 524880, 58240, 5714, 600, 66, 8, 1, 608608000, 96342400, 11022480, 1106560, 95064

Offset

0,4

Comments

This is the triple factorial variant of Moessner's factorial array described by A125714 and also of the double factorial array A135876. Another very interesting variant is A136217.

Links

Table of n, a(n) for n=0..49.

Formula

Columns 0, 1 and 2 form the triple factorials A007559, A008544 and A032031, respectively. Column 4 equals A024216; column 6 equals A024395.

Example

Square array begins:
(1),(1),(1),1,(1),1,(1),1,(1),1,1,(1),1,1,(1),1,1,(1),1,1,1,(1),1,1,1,...;
(1),(2),(3),4,(5),6,(7),8,(9),10,11,(12),13,14,(15),16,17,(18),19,20,21,..;
(4),(10),(18),28,(39),52,(66),82,(99),118,138,(159),182,206,(231),258,286,..;
(28),(80),(162),280,(418),600,(806),1064,(1350),1696,2074,(2485),2966,3484,..;
(280),(880),(1944),3640,(5714),8680,(12164),16840,(22194),29080,36824,(45474),.;
(3640),(12320),(29160),58240,(95064),151200,(219108),315440,(428652),581680,...;
(58240),(209440),(524880),1106560,(1864456),3082240,...;
where terms in parenthesis are at positions {[m*(m+5)/6], m>=0}
and are removed before taking partial sums to obtain the next row.
To generate the array, start with all 1's in row 0; from then on,
obtain row n+1 from row n by first removing terms in row n at
positions {[m*(m+5)/6], m>=0} and then taking partial sums.
For example, to generate row 2 from row 1:
[(1),(2),(3),4,(5),6,(7),8,(9),10,11,(12),13,14,(15),16,17,(18),...],
remove terms at positions [0,1,2,4,6,8,11,14,17,...] to get:
[4, 6, 8, 10,11, 13,14, 16,17, 19,20,21, 23,24,25, 27,28,29, ...]
then take partial sums to obtain row 2:
[4, 10, 18, 28,39, 52,66, 82,99, 118,138,159, 182,206,231, ...].
Continuing in this way will generate all the rows of this array.

Mathematica

t[n_, k_] := t[n, k] = Module[{a = 0, m = 0, c = 0, d = 0}, If[n == 0, a = 1, While[d <= k, If[c == Quotient[(m*(m + 5)), 6], m += 1, a += t[n - 1, c]; d += 1]; c += 1]]; a]; Table[t[n - k, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 06 2013, translated from Pari *)

Prog

(PARI) {T(n, k)=local(A=0, m=0, c=0, d=0); if(n==0, A=1, until(d>k, if(c==(m*(m+5))\6, m+=1, A+=T(n-1, c); d+=1); c+=1)); A}

Crossrefs

Cf. columns: A007559, A008544, A032031, A024216, A024395; related tables: A136213, A125714, A135876, A127054, A125781; A136217.

Sequence in context: A236961 A245958 A138271 * A136215 A136737 A275595

Adjacent sequences: A136209 A136210 A136211 * A136213 A136214 A136215

Keyword

nice,nonn,tabl

Author

Paul D. Hanna, Dec 22 2007

Status

approved