A335602 Number of 3-regular cubic partitions of n.

1, 1, 3, 3, 8, 9, 17, 20, 36, 43, 70, 84, 131, 158, 234, 284, 408, 495, 690, 837, 1143, 1385, 1852, 2241, 2952, 3565, 4626, 5574, 7150, 8595, 10903, 13074, 16434, 19656, 24494, 29223, 36146, 43016, 52836, 62722, 76572, 90675, 110063, 130021, 157014, 185049, 222388

Offset

0,3

Links

Table of n, a(n) for n=0..46.

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

H.-C. 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.

D. S. Gireesh and M. S. Mahadeva Naika, General family of congruences modulo large powers of 3 for cubic partition pairs, New Zealand J. Math. 47 (2017), 43--56.

Formula

G.f.: (f_3(x)*f_6(x)) / (f_1(x)*f_2(x)) where f_k(x) = Product_{m>=1} (1 - x^(m*k)).

a(n) ~ exp(sqrt(2*n/3)*Pi) / (6^(5/4) * n^(3/4)). - Vaclav Kotesovec, Jun 23 2020

Mathematica

nmax = 20; CoefficientList[Series[Product[(1 - x^(3*k)) * (1 - x^(6*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^3 + A)*eta(x^6 + A)/(eta(x + A)*eta(x^2 + A)))} \\ Andrew Howroyd, Jul 29 2020

Crossrefs

Cf. A002513, A335604.

Sequence in context: A135477 A182473 A363125 * A092549 A260890 A022663

Adjacent sequences: A335599 A335600 A335601 * A335603 A335604 A335605

Keyword

nonn

Author

Chandrappa Shivashankar, Jun 15 2020

Status

approved