A285927 - OEIS

A285927

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

%I #21 Oct 24 2024 12:33:35

%S 1,3,9,19,42,81,155,276,486,821,1368,2214,3541,5544,8586,13082,19740,

%T 29403,43414,63423,91935,132075,188418,266733,375232,524331,728514,

%U 1006216,1382604,1889739,2570719,3480420,4691682,6297102,8418252,11209347,14870970

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

%H Seiichi Manyama, <a href="/A285927/b285927.txt">Table of n, a(n) for n = 0..10000</a>

%H Mohammed L. Nadji and Moussa Ahmia, <a href="https://math.colgate.edu/~integers/y86/y86.pdf">Congruences for L-regular tripartitions for L in {2, 3}</a>, Integers (2024) Vol. 24, Art. No. A86. See p. 2.

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

%F a(n) ~ exp(2*Pi*sqrt(n/3)) / (2 * 3^(7/4) * n^(3/4)). - _Vaclav Kotesovec_, Apr 30 2017

%t 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 *)

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

%Y Cf. A000726, A199659.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Apr 28 2017