OFFSET
0,2
COMMENTS
Also the number of integer solutions (a_1, a_2, ..., a_n) to the equation a_1^2 + 2*a_2^2 + ... + n*a_n^2 = n.
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Seiichi Manyama)
Eric Weisstein's World of Mathematics, Jacobi Theta Functions
FORMULA
Expansion of Product_{k>0} eta(q^(2*k))^5 / (eta(q^k)*eta(q^(4*k)))^2.
a(n) ~ log(2)^(3/8) * exp(Pi*sqrt(n*log(2))) / (4 * Pi^(1/4) * n^(7/8)). - Vaclav Kotesovec, Oct 05 2018
Expansion of Product_{k>0} theta_4(q^(2*k))/theta_4(q^(2*k-1)), where theta_4() is the Jacobi theta function. - Seiichi Manyama, Oct 26 2018
MATHEMATICA
nmax = 50; CoefficientList[Series[Product[EllipticTheta[3, 0, x^k], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 05 2018 *)
nmax = 50; CoefficientList[Series[Product[(1 - x^(k*j))*(1 + x^(k*j))^3/(1 + x^(2*k*j))^2, {k, 1, nmax}, {j, 1, Floor[nmax/k] + 1}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 05 2018 *)
PROG
(PARI) m=50; x='x+O('x^m); Vec(1/(prod(k=1, 2*m, prod(j=1, floor(2*m/k), (1 - x^(k*j))*(1 + x^(k*j))^3/(1 + x^(2*k*j))^2 )))) \\ G. C. Greubel, Oct 29 2018
(Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((&*[(&*[(1 - x^(k*j))*(1 + x^(k*j))^3/(1 + x^(2*k*j))^2: j in [1..Floor(2*m/k)]]): k in [1..2*m]]))); // G. C. Greubel, Oct 29 2018
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
Seiichi Manyama, Oct 05 2018
STATUS
approved