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
Keywords
Links
- 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.
Programs
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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 *) -
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
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