A337968 L.g.f.: -log( Sum_{n=-oo..+oo} (-p)^n * (p*x)^(n^2) ) = Sum_{n>=1} a(n) * x^n/n, where p = sqrt(3).

4, 16, 64, 136, 424, 1216, 2944, 7696, 24868, 66016, 187840, 568864, 1632232, 4868992, 14760064, 43243552, 129586312, 390119248, 1163757376, 3491600176, 10483526656, 31391806528, 94159476352, 282588457024, 847413793324, 2541979556704, 7627065915520, 22877327084608, 68630932189288, 205904938577536

Offset

1,1

Links

Paul D. Hanna, Table of n, a(n) for n = 1..500

Formula

L.g.f.: -log( Sum_{n>=0} (-1)^n * A337969(n) * x^(n^2) ) = Sum_{n>=1} a(n) * x^n/n, where A337969(n) = 3^(n*(n-1)/2) + 3^(n*(n+1)/2) for n>0 with A337949(0) = 1.

L.g.f.: -log( Product_{n>=1} (1 - 3^n*x^(2*n)) * (1 - 3^n*x^(2*n-1)) * (1 - 3^(n-1)*x^(2*n-1)) ) = Sum_{n>=1} a(n) * x^n/n, by the Jacobi triple product identity.

L.g.f.: Sum_{n>=1} Sum_{k>=1} ( 3^(k*n)*x^(2*k*n) + (3^n+1)*3^((k-1)*n)*x^((2*k-1)*n) )/n = Sum_{n>=1} a(n) * x^n/n.

a(n) = 4 (mod 12) for n >= 1.

Example

L.g.f.: L(x) = 4*x + 16*x^2/2 + 64*x^3/3 + 136*x^4/4 + 424*x^5/5 + 1216*x^6/6 + 2944*x^7/7 + 7696*x^8/8 + 24868*x^9/9 + 66016*x^10/10 + 187840*x^11/11 + 568864*x^12/12 + 1632232*x^13/13 + 4868992*x^14/14 + 14760064*x^15/15 + 43243552*x^16/16 + ... + a(n)*x^n/n + ...
where
exp(-L(x)) = 1 - 4*x + 30*x^4 - 756*x^9 + 59778*x^16 - 14407956*x^25 + 10474702110*x^36 - 22887252808164*x^49 + ... + (-1)^n*A337969(n)*x^(n^2) + ...

Prog

(PARI) /* By Definition: */
{a(n) = n*polcoeff( -log( sum(m=-sqrtint(2*n+1), sqrtint(2*n+1), (-1)^m*3^(m*(m+1)/2)*x^(m^2) +x*O(x^n)) ), n)}
for(n=1, 30, print1(a(n), ", "))
(PARI) /* By the Jacobi Triple Product identity: */
{a(n) = n*polcoeff( -log( prod(m=1, n\2+1, (1 - 3^m*x^(2*m)) * (1 - 3^m*x^(2*m-1)) * (1 - 3^(m-1)*x^(2*m-1)) +x*O(x^n))), n)}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A337969, A337948, A337950.

Sequence in context: A275123 A275217 A158988 * A328850 A330687 A027676

Adjacent sequences: A337965 A337966 A337967 * A337969 A337970 A337971

Keyword

nonn

Author

Paul D. Hanna, Oct 04 2020

Status

approved