|
%I #9 Nov 19 2013 04:00:49
%S 1,2,4,14,90,438,3151,24390,204156,1833212,17301306,175936764,
%T 1870247133,20872753540,243478609605,2957875659062,37319273049382,
%U 487266892836348,6574891059415183,91475580555526776,1309960647920094337,19278546942842385994,291167370195970990704,4507447478297070537800
%N a(n) = [x^(n*(n+1)/2)] G(x)^(n+1) where G(x) = Sum_{n>=0} x^(n*(n+1)/2).
%H Paul D. Hanna, <a href="/A232108/b232108.txt">Table of n, a(n) for n = 0..100</a>
%e Let G(x) = 1 + x + x^3 + x^6 + x^10 + x^15 + x^21 + x^28 + x^36 +...
%e then a(n) = the coefficient of x^(n*(n+1)/2) in G(x)^n.
%e Coefficients of x^k in powers of G(x)^n begin:
%e n\k...0...1..2..3..4..5...6...7...8...9..10..11..12...13..14...15...16...
%e n=1: [(1),1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0,...];
%e n=2: [1, (2),1, 2, 2, 0, 3, 2, 0, 2, 2, 2, 1, 2, 0, 2, 4,...];
%e n=3: [1, 3, 3,(4),6, 3, 6, 9, 3, 7, 9, 6, 9, 9, 6, 6, 15,...];
%e n=4: [1, 4, 6, 8,13,12,(14),24, 18, 20, 32, 24, 31, 40, 30, 32, 48,...];
%e n=5: [1, 5,10,15,25,31, 35, 55, 60, 60,(90),90, 95, 135,125, 126, 170,...];
%e n=6: [1, 6,15,26,45,66, 82,120,156,170,231,276,290, 390,435,(438),561,...]; ...
%e the coefficients in parenthesis form the initial terms of this sequence.
%o (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)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A196010.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Nov 18 2013
|
