VOOZH about

URL: https://oeis.org/A337950

⇱ A337950 - OEIS

login
A337950
L.g.f.: -log( Sum_{n=-oo..+oo} (-2)^n * (2*x)^(n^2) ) = Sum_{n>=1} a(n) * x^n/n.
5
5, 25, 125, 353, 1425, 5425, 18625, 69121, 286145, 1082625, 4250625, 17072897, 67375105, 269185025, 1079450625, 4296933377, 17185439745, 68786663425, 274902810625, 1099633590273, 4399081242625, 17592482791425, 70369226522625, 281488801333249, 1125907377946625, 4503605214183425, 18014623543066625, 72057637105041409
OFFSET
1,1
LINKS
FORMULA
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. .
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.
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.
a(4*n + k) = 0 (mod 5) for n >= 0, and k = 1,2,3 (conjecture).
EXAMPLE
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 + ...
where
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) + ...
PROG
(PARI) /* By Definition: */
{a(n) = n*polcoeff( -log( sum(m=-sqrtint(n+1), sqrtint(n+1), (-2)^m*(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 - 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)}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
Sequence in context: A036149 A061974 A015950 * A366323 A267780 A228736
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 03 2020
STATUS
approved