A327047 Expansion of Product_{k>=1} (1 + x^k) * (1 + x^(2*k)) * (1 + x^(3*k)) * (1 + x^(4*k)) * (1 + x^(5*k)).

1, 1, 2, 4, 6, 10, 16, 23, 34, 51, 72, 101, 143, 195, 267, 366, 487, 650, 866, 1135, 1487, 1940, 2504, 3226, 4145, 5283, 6714, 8513, 10725, 13481, 16905, 21085, 26244, 32588, 40299, 49732, 61229, 75131, 92004, 112435, 137009, 166627, 202269, 244919, 296038

Offset: 0

Author

Vaclav Kotesovec, Aug 16 2019

Keywords

• nonn

Comments

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

Links

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

Crossrefs

Cf. A000009, A001935, A327045, A327046.

Cf. A107742, A327044.

Programs

Mathematica

nmax = 50; CoefficientList[Series[Product[(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^(1/4) * exp(sqrt(137*n/5)*Pi/6) / (2^(9/2)*sqrt(3)*5^(1/4)*n^(3/4)).