A177155 G.f.: exp( Integral (theta_3(x)^8-1)/(16x) dx ), where theta_3(x) = 1 + Sum_{n>=1} 2*x^(n^2) is a Jacobi theta function.

1, 1, 4, 13, 35, 87, 217, 539, 1291, 2999, 6880, 15595, 34738, 76202, 165282, 354655, 752546, 1580514, 3289337, 6787085, 13887937, 28195434, 56824772, 113729640, 226104615, 446665922, 877063515, 1712252521, 3324245063, 6419561961

Offset

0,3

Comments

Compare to g.f. of partitions in which no parts are multiples of 4:

g.f. of A001935 = exp( Integral (theta_3(x)^4-1)/(8x) dx ).

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Formula

G.f.: exp( Sum_{n>=1} A008457(n)*x^n/n ) where A008457(n) = Sum_{d|n} (-1)^(n-d)*d^3.

a(n) ~ exp(2*Pi*n^(3/4)/3 - Zeta(3)/Pi^2) / (4*n^(5/8)). - Vaclav Kotesovec, Apr 10 2019

Example

G.f.: A(x) = 1 + x + 4*x^2 + 13*x^3 + 35*x^4 + 87*x^5 +...
log(A(x)) = x + 7*x^2/2 + 28*x^3/3 + 71*x^4/4 + 126*x^5/5 +...+ A008457(n)*x^n/n +...

Mathematica

nmax = 40; Abs[CoefficientList[Series[Product[1/(1 - x^k)^((-1)^k*k^2), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Apr 10 2019 *)
nmax = 40; CoefficientList[Series[Product[(1 + x^(2*k - 1))^((2*k - 1)^2)/(1 - x^(2*k))^(4*k^2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 10 2019 *)

Prog

(PARI) {a(n)=polcoeff(exp(sum(m=1, n, sumdiv(m, d, (-1)^(m-d)*d^3)*x^m/m)+x*O(x^n)), n)}
(PARI) {a(n)=local(theta3=1+sum(m=1, sqrtint(2*n+2), 2*x^(m^2)+x*O(x^n))); polcoeff(exp(intformal((theta3^8-1)/(16*x))), n)}

Crossrefs

Cf. A138503, A008457, A001935, A000143, A307462.

Sequence in context: A266357 A095941 A210843 * A189595 A189602 A317781

Adjacent sequences: A177152 A177153 A177154 * A177156 A177157 A177158

Keyword

nonn

Author

Paul D. Hanna, May 03 2010, May 08 2010

Status

approved