A360191 G.f. 1 / Product_{n>=1} (1 - x^n)^3 * (1 - x^(2*n-1))^2.

1, 5, 18, 55, 149, 371, 867, 1923, 4086, 8374, 16634, 32152, 60669, 112041, 202943, 361200, 632647, 1091917, 1859225, 3126242, 5195715, 8541624, 13899866, 22404091, 35787815, 56683294, 89061028, 138872410, 214984454, 330532633, 504869316, 766357010, 1156355165

Offset

0,2

Comments

Self-convolution inverse of A080332.

Links

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

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following.

(!) A(x) = 1 / [Product_{n>=1} (1 - x^n)^3 * (1 - x^(2*n-1))^2].

(2) A(x) = 1 / [Sum_{n=-oo..+oo} (6*n + 1) * x^(n*(3*n + 1)/2)].

a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (12*sqrt(2)*n^(3/2)). - Vaclav Kotesovec, Feb 07 2023

Example

G.f.: A(x) = 1 + 5*x + 18*x^2 + 55*x^3 + 149*x^4 + 371*x^5 + 867*x^6 + 1923*x^7 + 4086*x^8 + 8374*x^9 + 16634*x^10 + 32152*x^11 + 60669*x^12 + ...

Mathematica

nmax = 30; CoefficientList[Series[1/Product[(1 - x^k)^3 * (1 - x^(2*k-1))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 07 2023 *)
nmax = 30; CoefficientList[Series[1/(QPochhammer[x] * EllipticTheta[4, 0, x]^2), {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 07 2023 *)

Prog

(PARI) {a(n) = polcoeff( 1/prod(m=1, n, (1 - x^m)^3 * (1 - x^(2*m-1))^2 +x*O(x^n)), n)}
for(n=0, 32, print1(a(n), ", "))

Crossrefs

Cf. A080332, A361535, A361050, A361550.

Sequence in context: A263318 A011845 A099450 * A344847 A145129 A001793

Adjacent sequences: A360188 A360189 A360190 * A360192 A360193 A360194

Keyword

nonn

Author

Paul D. Hanna, Jan 29 2023

Status

approved