A318027 Expansion of Product_{k>=1} 1/((1 - x^k)*(1 - x^(4*k))).

1, 1, 2, 3, 6, 8, 13, 18, 29, 39, 57, 77, 112, 148, 205, 271, 372, 484, 647, 838, 1110, 1423, 1852, 2361, 3051, 3857, 4922, 6191, 7849, 9805, 12319, 15314, 19131, 23649, 29333, 36099, 44556, 54568, 66963, 81683, 99803, 121229, 147413, 178411, 216111, 260590, 314365, 377819, 454229

Offset

0,3

Comments

Convolution of A000041 and A035444.

Convolution of A000712 and A082303.

Convolution inverse of A107034.

Number of partitions of n if there are 2 kinds of parts that are multiples of 4.

Links

Table of n, a(n) for n=0..48.

Zakir Ahmed, Nayandeep Deka Baruah, Manosij Ghosh Dastidar, New congruences modulo 5 for the number of 2-color partitions, Journal of Number Theory, Volume 157, December 2015, Pages 184-198.

Index entries for sequences related to partitions

Formula

G.f.: exp(Sum_{k>=1} x^k*(1 + x^k + x^(2*k) + 2*x^(3*k))/(k*(1 - x^(4*k)))).

a(n) ~ 5^(3/4) * exp(sqrt(5*n/6)*Pi) / (2^(13/4) * 3^(3/4) * n^(5/4)). - Vaclav Kotesovec, Aug 14 2018

Example

a(5) = 8 because we have [5], [4, 1], [4', 1], [3, 2], [3, 1, 1], [2, 2, 1], [2, 1, 1, 1] and [1, 1, 1, 1, 1].

Maple

a:=series(mul(1/((1-x^k)*(1-x^(4*k))), k=1..55), x=0, 49): seq(coeff(a, x, n), n=0..48); # Paolo P. Lava, Apr 02 2019

Mathematica

nmax = 48; CoefficientList[Series[Product[1/((1 - x^k) (1 - x^(4 k))), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 48; CoefficientList[Series[1/(QPochhammer[x] QPochhammer[x^4]), {x, 0, nmax}], x]
nmax = 48; CoefficientList[Series[Exp[Sum[x^k (1 + x^k + x^(2 k) + 2*x^(3 k))/(k (1 - x^(4 k))), {k, 1, nmax}]], {x, 0, nmax}], x]
Table[Sum[PartitionsP[k] PartitionsP[n - 4 k], {k, 0, n/4}], {n, 0, 48}]

Crossrefs

Cf. A000041, A000712, A002512 (self-convolution), A002513, A035444, A082303, A100853, A107034, A318026, A318028.

Sequence in context: A226635 A024788 A285472 * A373295 A004101 A003405

Adjacent sequences: A318024 A318025 A318026 * A318028 A318029 A318030

Keyword

nonn

Author

Ilya Gutkovskiy, Aug 13 2018

Status

approved