A337948 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(2).

3, 9, 27, 41, 93, 189, 297, 481, 1161, 1809, 3105, 6449, 10689, 20673, 44577, 73217, 144129, 299457, 553473, 1107201, 2243457, 4299777, 8529921, 17203969, 34030593, 67604481, 136001025, 269709313, 538296321, 1081023489, 2150531073, 4299030529, 8612255745, 17190158337, 34391638017, 68800294913

Offset

1,1

Links

Paul D. Hanna, Table of n, a(n) for n = 1..900

Formula

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

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

a(2*n+k) = 0 (mod 3) for n >= 0, and k = 1,2,3 (conjecture).

Example

L.g.f.: L(x) = 3*x + 9*x^2/2 + 27*x^3/3 + 41*x^4/4 + 93*x^5/5 + 189*x^6/6 + 297*x^7/7 + 481*x^8/8 + 1161*x^9/9 + 1809*x^10/10 + 3105*x^11/11 + 6449*x^12/12 + 10689*x^13/13 + 20673*x^14/14 + 44577*x^15/15 + 73217*x^16/16 + ... + a(n)*x^n/n + ...
where
exp(-L(x)) = 1 - 3*x + 10*x^4 - 72*x^9 + 1088*x^16 - 33792*x^25 + 2129920*x^36 - 270532608*x^49 + 68987912192*x^64 + ... + (-1)^n*A337949(n)*x^(n^2) + ...

Prog

(PARI) /* By Definition: */
{a(n) = n*polcoeff( -log( sum(m=-sqrtint(n+1), sqrtint(n+1), (-1)^m*2^(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 - 2^m*x^(2*m)) * (1 - 2^m*x^(2*m-1)) * (1 - 2^(m-1)*x^(2*m-1)) +x*O(x^n))), n)}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A337949, A337950.

Sequence in context: A070361 A056024 A116475 * A163791 A248078 A057829

Adjacent sequences: A337945 A337946 A337947 * A337949 A337950 A337951

Keyword

nonn

Author

Paul D. Hanna, Oct 03 2020

Status

approved