A285927 Expansion of (Product_{k>0} (1 - x^(3*k)) / (1 - x^k))^3 in powers of x.

1, 3, 9, 19, 42, 81, 155, 276, 486, 821, 1368, 2214, 3541, 5544, 8586, 13082, 19740, 29403, 43414, 63423, 91935, 132075, 188418, 266733, 375232, 524331, 728514, 1006216, 1382604, 1889739, 2570719, 3480420, 4691682, 6297102, 8418252, 11209347, 14870970

Offset: 0

Author

Seiichi Manyama, Apr 28 2017

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• Mohammed L. Nadji and Moussa Ahmia, Congruences for L-regular tripartitions for L in {2, 3}, Integers (2024) Vol. 24, Art. No. A86. See p. 2.

Crossrefs

(Product_{k>0} (1 - x^(m*k)) / (1 - x^k))^m: A022567 (m=2), this sequence (m=3), A093160 (m=4), A285928 (m=5).

Cf. A000726, A199659.

Programs

Mathematica

nmax = 40; CoefficientList[Series[Product[((1 - x^(3*k)) / (1 - x^k))^3, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 30 2017 *)

Formula

a(0) = 1, a(n) = (3/n)*Sum_{k=1..n} A046913(k)*a(n-k) for n > 0.

a(n) ~ exp(2*Pi*sqrt(n/3)) / (2 * 3^(7/4) * n^(3/4)). - Vaclav Kotesovec, Apr 30 2017