A197707 G.f.: A(x) = 1 + Sum_{n>=1} x^(n^2) * ((1-x)^n + 1/(1-x)^n).

1, 2, 0, 1, 3, 1, 5, 5, 6, 9, 8, 18, 19, 26, 33, 41, 52, 60, 87, 99, 132, 166, 209, 261, 323, 398, 481, 604, 716, 893, 1086, 1331, 1629, 1991, 2428, 2952, 3578, 4314, 5217, 6229, 7508, 8967, 10737, 12838, 15345, 18334, 21894, 26127, 31149, 37093, 44100

Offset

0,2

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Formula

G.f.: A(x) = Product_{n>=1} (1 - x^(2*n)) * (1 + x^(2*n-1)*(1-x)) * (1 + x^(2*n-1)/(1-x)), due to the Jacobi triple product identity.

Example

G.f.: A(x) = 1 + 2*x + x^3 + 3*x^4 + x^5 + 5*x^6 + 5*x^7 + 6*x^8 +...
where the g.f. A(x) may be expressed as the q-series:
A(x) = 1 + x*((1-x) + 1/(1-x)) + x^4*((1-x)^2 + 1/(1-x)^2) + x^9*((1-x)^3 + 1/(1-x)^3) + x^16*((1-x)^4 + 1/(1-x)^4) +...
and the Jacobi triple product:
A(x) = (1-x^2)*(1+x*(1-x))*(1+x/(1-x)) * (1-x^4)*(1+x^3*(1-x))*(1+x^3/(1-x)) * (1-x^6)*(1+x^5*(1-x))*(1+x^5/(1-x)) *...

Prog

(PARI) {a(n)=local(A); A=1+sum(m=1, sqrtint(n)+1, x^(m^2)*((1-x)^m+1/(1-x+x*O(x^n))^m)); polcoeff(A, n)}
(PARI) {a(n)=local(A); A=prod(m=1, n\2+1, (1-x^(2*m))*(1+x^(2*m-1)*(1-x))*(1+x^(2*m-1)/(1-x+x*O(x^n)))); polcoeff(A, n)}

Crossrefs

Cf. A190791.

Sequence in context: A029274 A239498 A079219 * A253668 A216220 A334892

Adjacent sequences: A197704 A197705 A197706 * A197708 A197709 A197710

Keyword

nonn

Author

Paul D. Hanna, Oct 17 2011

Status

approved