0,2

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).

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.

(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))}

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

Sequence in context: A026072 A063646 A047730 this_sequence A059040 A074530 A159662

Adjacent sequences: A162432 A162433 A162434 this_sequence A162436 A162437 A162438

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Jul 03 2009