A337968 - OEIS

%I #9 Oct 06 2020 19:46:37

%S 4,16,64,136,424,1216,2944,7696,24868,66016,187840,568864,1632232,

%T 4868992,14760064,43243552,129586312,390119248,1163757376,3491600176,

%U 10483526656,31391806528,94159476352,282588457024,847413793324,2541979556704,7627065915520,22877327084608,68630932189288,205904938577536

%N 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).

%H Paul D. Hanna, <a href="/A337968/b337968.txt">Table of n, a(n) for n = 1..500</a>

%F 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.

%F 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.

%F 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.

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

%e 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 + ...

%e where

%e 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) + ...

%o (PARI) /* By Definition: */

%o {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)}

%o for(n=1,30,print1(a(n),", "))

%o (PARI) /* By the Jacobi Triple Product identity: */

%o {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)}

%o for(n=1,30,print1(a(n),", "))

%Y Cf. A337969, A337948, A337950.

%K nonn

%O 1,1

%A _Paul D. Hanna_, Oct 04 2020