A177155 - OEIS

%I #10 Apr 10 2019 03:18:26

%S 1,1,4,13,35,87,217,539,1291,2999,6880,15595,34738,76202,165282,

%T 354655,752546,1580514,3289337,6787085,13887937,28195434,56824772,

%U 113729640,226104615,446665922,877063515,1712252521,3324245063,6419561961

%N 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.

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

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

%H Seiichi Manyama, <a href="/A177155/b177155.txt">Table of n, a(n) for n = 0..10000</a>

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

%F a(n) ~ exp(2*Pi*n^(3/4)/3 - Zeta(3)/Pi^2) / (4*n^(5/8)). - _Vaclav Kotesovec_, Apr 10 2019

%e G.f.: A(x) = 1 + x + 4*x^2 + 13*x^3 + 35*x^4 + 87*x^5 +...

%e 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 +...

%t 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 *)

%t 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 *)

%o (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)}

%o (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)}

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

%K nonn

%O 0,3

%A _Paul D. Hanna_, May 03 2010, May 08 2010