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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. 3
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified October 14 12:31 EDT 2019. Contains 328006 sequences. (Running on oeis4.)