A327044 Expansion of Product_{k>=1} 1/((1 - x^k) * (1 - x^(2*k)) * (1 - x^(3*k)) * (1 - x^(4*k)) * (1 - x^(5*k))).

1, 1, 3, 5, 11, 17, 33, 50, 89, 135, 223, 332, 530, 775, 1190, 1724, 2576, 3677, 5380, 7586, 10895, 15203, 21480, 29666, 41373, 56593, 77965, 105755, 144155, 193947, 261894, 349719, 468193, 620910, 824743, 1086661, 1433205, 1876865, 2459100, 3202155, 4170043

Offset: 0

Author

Vaclav Kotesovec, Aug 16 2019

Keywords

• nonn

Comments

Differs from A006170.

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

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000041, A002513, A327042, A327043.

Cf. A006171.

Programs

Mathematica

nmax = 50; CoefficientList[Series[Product[1/((1 - x^k) * (1 - x^(2*k)) * (1 - x^(3*k)) * (1 - x^(4*k)) * (1 - x^(5*k))), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ 137^(3/2) * exp(sqrt(137*n/10)*Pi/3) / (2880*sqrt(6)*n^2).