A337969 a(n) = 3^(n*(n-1)/2) + 3^(n*(n+1)/2) for n > 0, with a(0) = 1.

1, 4, 30, 756, 59778, 14407956, 10474702110, 22887252808164, 150117512089454082, 2954462801186130697764, 174452165321826729904869150, 30903328831843621482099712974036, 16423234171415040778843829721351058818, 26183907127466405538332338669897982729379636

Offset

0,2

Links

Table of n, a(n) for n=0..13.

Formula

G.f.: Sum_{n=-oo..+oo} 3^(n*(n+1)/2) * x^(n^2) = Sum_{n>=0} a(n) * x^(n^2).

G.f.: 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>=0} a(n) * x^(n^2), by the Jacobi triple product identity.

Example

G.f.: A(x) = 1 + 4*x + 30*x^4 + 756*x^9 + 59778*x^16 + 14407956*x^25 + 10474702110*x^36 + 22887252808164*x^49 + 150117512089454082*x^64 + 2954462801186130697764*x^81 + ... + a(n)*x^(n^2) + ...
which can be generated by the Jacobi Triple Product:
A(x) = (1 - 3*x^2)*(1 + 3*x)*(1 + x) * (1 - 3^2*x^4)*(1 + 3^2*x^3)*(1 + 3*x^3) * (1 - 3^3*x^6)*(1 + 3^3*x^5)*(1 + 3^2*x^5) * (1 - 3^4*x^8)*(1 + 3^4*x^7)*(1 + 3^3*x^7) * ... * (1 - 3^n*x^(2*n))*(1 + 3^n*x^(2*n-1))*(1 + 3^(n-1)*x^(2*n-1)) * ...

Prog

(PARI) {a(n) = if(n==0, 1, 3^(n*(n-1)/2) + 3^(n*(n+1)/2))}
for(n=0, 15, print1(a(n), ", "))
(PARI) /* As Coefficients in a Jacobi Theta Function: */
{a(n) = polcoeff( sum(m=-n, n, 3^(m*(m+1)/2)*x^(m^2) +x*O(x^(n^2))), n^2)}
for(n=0, 15, print1(a(n), ", "))
(PARI) /* By the Jacobi Triple Product identity: */
{a(n) = polcoeff( prod(m=1, n^2\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^2))), n^2)}
for(n=0, 15, print1(a(n), ", "))

Crossrefs

Cf. A337968, A337949, A337951.

Sequence in context: A120338 A064050 A162128 * A234566 A365366 A201974

Adjacent sequences: A337966 A337967 A337968 * A337970 A337971 A337972

Keyword

nonn

Author

Paul D. Hanna, Oct 04 2020

Status

approved