A337970 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(5).

6, 36, 216, 776, 3876, 18576, 83376, 399376, 2034126, 9884376, 49034376, 245689376, 1221921876, 6107609376, 30563759376, 152599609376, 762979296876, 3815571921876, 19073708984376, 95369005859376, 476858422109376, 2384189755859376, 11920935693359376, 59605116212109376, 298023413134765626

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 * A337971(n) * x^(n^2) ) = Sum_{n>=1} a(n) * x^n/n, where A337971(n) = 5^(n*(n-1)/2) + 5^(n*(n+1)/2) for n>0 with A337971(0) = 1.

L.g.f.: -log( Product_{n>=1} (1 - 5^n*x^(2*n)) * (1 - 5^n*x^(2*n-1)) * (1 - 5^(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} ( 5^(k*n)*x^(2*k*n) + (5^n+1)*5^((k-1)*n)*x^((2*k-1)*n) )/n = Sum_{n>=1} a(n) * x^n/n.

a(n) = 6 (mod 10).

Example

L.g.f.: L(x) = 6*x + 36*x^2/2 + 216*x^3/3 + 776*x^4/4 + 3876*x^5/5 + 18576*x^6/6 + 83376*x^7/7 + 399376*x^8/8 + 2034126*x^9/9 + 9884376*x^10/10 + 49034376*x^11/11 + 245689376*x^12/12 + 1221921876*x^13/13 + 6107609376*x^14/14 + 30563759376*x^15/15 + 152599609376*x^16/16 + ... + a(n)*x^n/n + ...
where
exp(-L(x)) = 1 - 6*x + 130*x^4 - 15750*x^9 + 9781250*x^16 - 30527343750*x^25 + 476867675781250*x^36 + ... + (-1)^n*A337971(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*5^(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 - 5^m*x^(2*m)) * (1 - 5^m*x^(2*m-1)) * (1 - 5^(m-1)*x^(2*m-1)) +x*O(x^n))), n)}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A337971, A337948, A337950, A337968.

Sequence in context: A036162 A061708 A216127 * A124535 A267789 A228737

Adjacent sequences: A337967 A337968 A337969 * A337971 A337972 A337973

Keyword

nonn

Author

Paul D. Hanna, Oct 04 2020

Status

approved