A162434 Antidiagonal sums of table A162430.

1, 2, 5, 12, 32, 82, 218, 598, 1702, 4925, 14496, 43720, 134854, 423988, 1357884, 4425384, 14664610, 49386081, 168884638, 586012116, 2061993475, 7353614415, 26566226591, 97177708497, 359765742126, 1347447717234, 5103618615825

Offset

1,2

Links

Paul D. Hanna, Table of n, a(n) for n = 1..60

Formula

a(n) = Sum_{k=0..n} Sum_{m=k(k+1)/2..k(k+1)/2+k} [x^m] S(x)^(n-k) for n>=1 where S(x) = Sum_{n>=0} x^(n(n+1)/2).

Example

Let coefficients in powers of the series:
S = 1 + x + x^3 + x^6 + x^10 + x^15 + x^21 + x^28 + x^36 +...
form the following sequences:
S^1: [(1),(1,0),(1,0,0),(1,0,0,0),(1,0,0,0,0),...]
S^2: [(1),(2,1),(2,2,0),(3,2,0,2),(2,2,1,2,0),...]
S^3: [(1),(3,3),(4,6,3),(6,9,3,7),(9,6,9,9,6),...]
S^4: [(1),(4,6),(8,13,12),(14,24,18,20),...]
S^5: [(1),(5,10),(15,25,31),(35,55,60,60),...]
S^6: [(1),(6,15),(26,45,66),(82,120,156,170),...]
...
then the sums of the above grouped terms (in parenthesis)
form the initial terms of the rows of table A162430:
1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,...
1,3,4,7,7,9,12,11,15,14,19,19,19,22,23,27,26,...
1,6,13,25,39,52,81,97,129,154,187,234,250,321,...
1,10,33,76,157,264,425,626,897,1230,1629,2174,...
1,15,71,210,535,1086,1965,3431,5425,8181,12165,...
1,21,137,528,1622,3921,8254,16396,29136,48773,...
1,28,245,1219,4494,12936,31767,70826,141891,...
1,36,414,2621,11602,39622,112951,283574,637706,...
...
The antidiagonal sums of the above table forms this sequence.

Mathematica

t[n_, k_] := Module[{s = Sum[x^(m*(m+1)/2), {m, 0, k+1}] + O[x]^((k+1)*(k+2)/2)}, Sum[Coefficient[s^n, x, m], {m, k*(k+1)/2, k*(k+1)/2+k}]]; Table[Sum[t[n-k+1, k], {k, 0, n}], {n, 0, 26}] (* Jean-François Alcover, Nov 18 2013 *)

Prog

(PARI) {a(n)=local(S=sum(m=0, n+1, x^(m*(m+1)/2))+O(x^((n+1)*(n+2)/2))); sum(k=0, n, sum(m=k*(k+1)/2, k*(k+1)/2+k, polcoeff(S^(n-k), m)))}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A162430, A162431, A162432, A162433, A162435, A162428 (variant).

Sequence in context: A232215 A265265 A293868 * A277863 A039809 A335456

Adjacent sequences: A162431 A162432 A162433 * A162435 A162436 A162437

Keyword

nonn

Author

Paul D. Hanna, Jul 03 2009

Status

approved