A335604 Number of 9-regular cubic partitions of n.

1, 1, 3, 4, 9, 12, 23, 31, 54, 72, 117, 156, 242, 320, 477, 628, 909, 1188, 1676, 2178, 3012, 3888, 5283, 6780, 9079, 11582, 15309, 19424, 25389, 32040, 41462, 52063, 66780, 83448, 106182, 132084, 166862, 206660, 259359, 319896, 399069, 490272, 608234, 744444, 918864

Offset

0,3

Links

Table of n, a(n) for n=0..44.

Hei-Chi Chan, Ramanujan's cubic continued fraction and a generalization of his "most beautiful identity", Int. J. Number Theory 6 (2010), 673--680.

Hei-Chi Chan, Ramanujan's cubic continued fraction and Ramanujan type congruences for a certain partition function, Int. J. Number Theory 6 (2010), 819--834.

S. Chern, Arithmetic Properties for Cubic Partition Pairs Modulo Powers of 3, Acta. Math. Sin.-English Ser. 2017 33: 1504.

Bernard L. S. Lin, Congruences modulo 27 for cubic partition pairs, J. Number Theory 171 (2017), 31--42.

Formula

G.f.: (f_9(x)*f_18(x)) / (f_1(x)*f_2(x)) where f_k(x) = Product_{m>=1} (1 - x^(m*k)).

a(n) ~ exp(2*sqrt(2*n)*Pi/3) / (2^(3/4) * 3^(5/2) * n^(3/4)). - Vaclav Kotesovec, Jun 23 2020

Mathematica

nmax = 50; CoefficientList[Series[Product[(1 - x^(9*k)) * (1 - x^(18*k)) / ((1 - x^k) * (1 - x^(2*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 23 2020 *)

Prog

(PARI) seq(n)={my(A=O(x*x^n)); Vec(eta(x^9 + A)*eta(x^18 + A)/(eta(x + A)*eta(x^2 + A)))} \\ Andrew Howroyd, Jul 29 2020

Crossrefs

Cf. A002513, A335602.

Sequence in context: A293569 A304825 A026476 * A002513 A034418 A034421

Adjacent sequences: A335601 A335602 A335603 * A335605 A335606 A335607

Keyword

nonn

Author

Chandrappa Shivashankar, Jun 15 2020

Status

approved