A285338 Expansion of Product_{k>=1} (1 + x^(5*k-4))^(5*k-4).

1, 1, 0, 0, 0, 0, 6, 6, 0, 0, 0, 11, 26, 15, 0, 0, 16, 82, 86, 20, 0, 21, 172, 316, 180, 15, 26, 328, 872, 790, 226, 37, 538, 2043, 2681, 1310, 202, 845, 4184, 7426, 5390, 1447, 1290, 7855, 18067, 17705, 7277, 2662, 13723, 39468, 50030, 28707, 8742, 22979, 79760

Offset: 0

Author

Vaclav Kotesovec, Apr 17 2017

Keywords

• nonn

Comments

For all n<=30 a(n) = abs(A285071(n)), but a(31) <> abs(A285071(31)).

In general, if m >= 1 and g.f. = Product_{k>=1} (1 + x^(m*k-m+1))^(m*k-m+1), then a(n, m) ~ exp(2^(-4/3) * 3^(4/3) * m^(-1/3) * Zeta(3)^(1/3) * n^(2/3)) * Zeta(3)^(1/6) / (2^(1/6 + 1/(2*m) + m/12) * 3^(1/3) * m^(1/6) * sqrt(Pi) * n^(2/3)).

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vaclav Kotesovec)

Crossrefs

Product_{k>=0} (1 + x^(m*k+1))^(m*k+1): A026007 (m=1), A262736 (m=2), A262949 (m=3), A285288 (m=4), this sequence (m=5).

Cf. A285071, A285340.

Programs

Mathematica

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

Formula

a(n) ~ exp(2^(-4/3) * 3^(4/3) * 5^(-1/3) * Zeta(3)^(1/3) * n^(2/3)) * Zeta(3)^(1/6) / (2^(41/60) * 3^(1/3) * 5^(1/6) * sqrt(Pi) * n^(2/3)).