A261636 Expansion of Product_{k>=0} 1/(1-x^(4*k+1))^4.

1, 4, 10, 20, 35, 60, 100, 160, 245, 364, 536, 780, 1115, 1564, 2166, 2980, 4065, 5484, 7326, 9720, 12830, 16824, 21902, 28344, 36510, 46820, 59736, 75844, 95910, 120844, 151688, 189668, 236330, 293564, 363542, 448804, 552425, 678144, 830338, 1014052, 1235296

Offset

0,2

Comments

In general, if j > 0, a > 0, b > 0, GCD(a,b) = 1 and g.f. = Product_{k>=0} 1/(1 - x^(a*k+b))^j, then a(n) ~ Gamma(b/a)^j * 2^(-(j+5)/4 - j*b/(2*a)) * 3^((j-1)/4 - j*b/(2*a)) * j^(-(j-1)/4 + j*b/(2*a)) * a^(-(j+1)/4 + j*b/(2*a)) * Pi^(-j + j*b/a) * n^((j-3)/4 - j*b/(2*a)) * exp(Pi*sqrt(2*j*n/(3*a))).

Links

Table of n, a(n) for n=0..40.

Formula

a(n) ~ exp(Pi*sqrt(2*n/3)) * 6^(1/4) * Gamma(1/4)^4 / (32 * Pi^3 * n^(1/4)).

Mathematica

nmax=50; CoefficientList[Series[Product[1/(1-x^(4*k+1))^4, {k, 0, nmax}], {x, 0, nmax}], x]

Crossrefs

Cf. A035382, A261616, A261635, A035451, A261629, A261632.

Sequence in context: A373963 A137359 A134987 * A058539 A354001 A368174

Adjacent sequences: A261633 A261634 A261635 * A261637 A261638 A261639

Keyword

nonn

Author

Vaclav Kotesovec, Aug 27 2015

Status

approved