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%I #8 Oct 06 2020 19:43:51
%S 5,25,125,353,1425,5425,18625,69121,286145,1082625,4250625,17072897,
%T 67375105,269185025,1079450625,4296933377,17185439745,68786663425,
%U 274902810625,1099633590273,4399081242625,17592482791425,70369226522625,281488801333249,1125907377946625,4503605214183425,18014623543066625,72057637105041409
%N L.g.f.: -log( Sum_{n=-oo..+oo} (-2)^n * (2*x)^(n^2) ) = Sum_{n>=1} a(n) * x^n/n.
%H Paul D. Hanna, <a href="/A337950/b337950.txt">Table of n, a(n) for n = 1..500</a>
%F L.g.f.: -log( Sum_{n>=0} (-1)^n*A337951(n) * x^(n^2) ) = Sum_{n>=1} a(n) * x^n/n, where A337951(n) = 2^(n*(n-1)) + 2^(n*(n+1)) for n>0 with A337951(0) = 1. .
%F L.g.f.: -log( Product_{n>=1} (1 - 4^n*x^(2*n)) * (1 - 4^n*x^(2*n-1)) * (1 - 4^(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} ( 4^(k*n)*x^(2*k*n) + (4^n+1)*4^((k-1)*n)*x^((2*k-1)*n) )/n = Sum_{n>=1} a(n) * x^n/n.
%F a(4*n + k) = 0 (mod 5) for n >= 0, and k = 1,2,3 (conjecture).
%e L.g.f.: L(x) = 5*x + 25*x^2/2 + 125*x^3/3 + 353*x^4/4 + 1425*x^5/5 + 5425*x^6/6 + 18625*x^7/7 + 69121*x^8/8 + 286145*x^9/9 + 1082625*x^10/10 + 4250625*x^11/11 + 17072897*x^12/12 + 67375105*x^13/13 + 269185025*x^14/14 + 1079450625*x^15/15 + 4296933377*x^16/16 + ... + a(n)*x^n/n + ...
%e where
%e exp(-L(x)) = 1 - 5*x + 68*x^4 - 4160*x^9 + 1052672*x^16 - 1074790400*x^25 + 4399120252928*x^36 - 72061992084439040*x^49 + 4722438540463683141632*x^64 + ... + (-1)^n*A337951(n)*x^(n^2) + ...
%o (PARI) /* By Definition: */
%o {a(n) = n*polcoeff( -log( sum(m=-sqrtint(n+1),sqrtint(n+1), (-2)^m*(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 - 4^m*x^(2*m)) * (1 - 4^m*x^(2*m-1)) * (1 - 4^(m-1)*x^(2*m-1)) +x*O(x^n))),n)}
%o for(n=1,30,print1(a(n),", "))
%Y Cf. A337951, A337948.
%K nonn
%O 1,1
%A _Paul D. Hanna_, Oct 03 2020
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