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
Keywords
Links
- 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.
Programs
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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 *) -
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
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