A162435 Main diagonal of table A162430.

1, 3, 13, 76, 535, 3921, 31767, 283574, 2679558, 26954099, 285775908, 3183625080, 37069123612, 449410859046, 5658593670526, 73786800505502, 994254712074330, 13815743636007248, 197629567539080448, 2905687475271339406

Offset

0,2

Links

Table of n, a(n) for n=0..19.

Formula

a(n) = Sum_{m=n(n+1)/2..n(n+1)/2+n} [x^m] S(x)^(n+1) for n>=0 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 main diagonal 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[t[n+1, n], {n, 0, 19}] (* 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(m=n*(n+1)/2, n*(n+1)/2+n, polcoeff(S^(n+1), m))}

Crossrefs

Cf. A162430, A162431, A162432, A162433, A162434, A162429 (variant).

Sequence in context: A026072 A063646 A047730 * A059040 A220895 A352308

Adjacent sequences: A162432 A162433 A162434 * A162436 A162437 A162438

Keyword

nonn

Author

Paul D. Hanna, Jul 03 2009

Status

approved