A206623 G.f.: Product_{n>0} ( (1+x^n)/(1-x^n) )^(n^3).
1, 2, 18, 88, 398, 1768, 7508, 30644, 121310, 467234, 1756080, 6457168, 23274788, 82381584, 286760344, 982874120, 3320800590, 11070619228, 36446345198, 118581503192, 381552358872, 1214868568728, 3829841265428, 11959828895612, 37013411304892, 113570015855642
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + 2*x + 18*x^2 + 88*x^3 + 398*x^4 + 1768*x^5 + 7508*x^6 +... where A(x) = (1+x)/(1-x) * (1+x^2)^8/(1-x^2)^8 * (1+x^3)^27/(1-x^3)^27 *... Also, A(x) = Euler transform of [2,15,54,120,250,405,686,960,1458,...]: A(x) = 1/((1-x)^2*(1-x^2)^15*(1-x^3)^54*(1-x^4)^120*(1-x^5)^250*(1-x^6)^405*...).
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..4595 (terms 0..1000 from Vaclav Kotesovec)
- Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 23.
Programs
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Mathematica
nmax = 40; CoefficientList[Series[Product[((1+x^k)/(1-x^k))^(k^3), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 19 2015 *) -
PARI
{a(n)=polcoeff(prod(m=1,n+1,((1+x^m)/(1-x^m+x*O(x^n)))^(m^3)),n)} -
PARI
{a(n)=polcoeff(exp(sum(m=1, n, (sigma(2*m, 4)-sigma(m, 4))/8*x^m/m)+x*O(x^n)), n)} -
PARI
{a(n)=local(InvEulerGF=x*(2+15*x+46*x^2+60*x^3+46*x^4+15*x^5+2*x^6)/(1-x^2+x*O(x^n))^4);polcoeff(1/prod(k=1,n,(1-x^k+x*O(x^n))^polcoeff(InvEulerGF,k)),n)} for(n=0,30,print1(a(n),", "))
Formula
G.f.: exp( Sum_{n>=1} (sigma_4(2*n) - sigma_4(n))/8 * x^n/n ), where sigma_4(n) is the sum of 4th powers of divisors of n (A001159).
Inverse Euler transform has g.f.: x*(2 + 15*x + 46*x^2 + 60*x^3 + 46*x^4 + 15*x^5 + 2*x^6)/(1-x^2)^4.
a(n) ~ (93*Zeta(5))^(59/600) * exp(5/4 * (93*Zeta(5)/2)^(1/5) * n^(4/5) + Zeta'(-3)) / (2^(59/100) * sqrt(5*Pi) * n^(359/600)), where Zeta(5) = A013663, Zeta'(-3) = A259068. - Vaclav Kotesovec, Aug 19 2015
Comments