A232108 a(n) = [x^(n*(n+1)/2)] G(x)^(n+1) where G(x) = Sum_{n>=0} x^(n*(n+1)/2).

1, 2, 4, 14, 90, 438, 3151, 24390, 204156, 1833212, 17301306, 175936764, 1870247133, 20872753540, 243478609605, 2957875659062, 37319273049382, 487266892836348, 6574891059415183, 91475580555526776, 1309960647920094337, 19278546942842385994, 291167370195970990704, 4507447478297070537800

Offset: 0

Author

Paul D. Hanna, Nov 18 2013

Keywords

• nonn

Examples

			Let G(x) = 1 + x + x^3 + x^6 + x^10 + x^15 + x^21 + x^28 + x^36 +...
then a(n) = the coefficient of x^(n*(n+1)/2) in G(x)^n.
Coefficients of x^k in powers of G(x)^n begin:
n\k...0...1..2..3..4..5...6...7...8...9..10..11..12...13..14...15...16...
n=1: [(1),1, 0, 1, 0, 0,  1,  0,  0,  0,  1,  0,  0,   0,  0,   1,   0,...];
n=2: [1, (2),1, 2, 2, 0,  3,  2,  0,  2,  2,  2,  1,   2,  0,   2,   4,...];
n=3: [1,  3, 3,(4),6, 3,  6,  9,  3,  7,  9,  6,  9,   9,  6,   6,  15,...];
n=4: [1,  4, 6, 8,13,12,(14),24, 18, 20, 32, 24, 31,  40, 30,  32,  48,...];
n=5: [1,  5,10,15,25,31, 35, 55, 60, 60,(90),90, 95, 135,125, 126, 170,...];
n=6: [1,  6,15,26,45,66, 82,120,156,170,231,276,290, 390,435,(438),561,...]; ...
the coefficients in parenthesis form the initial terms of this sequence.
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..100

Crossrefs

Cf. A196010.

Programs

PARI

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