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A162435 Main diagonal of table A162430. 6
1, 3, 13, 76, 535, 3921, 31767, 283574, 2679558, 26954099, 285775908, 3183625080, 37069123612, 449410859046, 5658593670526, 73786800505502, 994254712074330, 13815743636007248, 197629567539080448, 2905687475271339406 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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 A336421

Adjacent sequences:  A162432 A162433 A162434 * A162436 A162437 A162438

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Jul 03 2009

STATUS

approved

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Last modified August 3 20:08 EDT 2020. Contains 336201 sequences. (Running on oeis4.)