A318028 Expansion of Product_{k>=1} 1/((1 - x^k)*(1 - x^(5*k))).

1, 1, 2, 3, 5, 8, 12, 17, 25, 35, 51, 69, 96, 129, 175, 235, 312, 410, 539, 700, 913, 1173, 1508, 1923, 2450, 3105, 3920, 4926, 6177, 7710, 9614, 11923, 14766, 18218, 22435, 27550, 33750, 41231, 50278, 61150, 74259, 89932, 108744, 131193, 158025, 189979, 227998, 273125, 326692

Offset

0,3

Comments

Convolution of A000712 and A145466.

Convolution inverse of A030202.

Number of partitions of n if there are 2 kinds of parts that are multiples of 5.

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) + x^(3*k) + 2*x^(4 k))/(k*(1 - x^(5*k)))).

a(n) ~ exp(2*Pi*sqrt(n/5)) / (4 * 5^(1/4) * n^(5/4)). - Vaclav Kotesovec, Aug 14 2018

Example

a(5) = 8 because we have [5], [5'], [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^(5*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^(5 k))), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 48; CoefficientList[Series[1/(QPochhammer[x] QPochhammer[x^5]), {x, 0, nmax}], x]
nmax = 48; CoefficientList[Series[Exp[Sum[x^k (1 + x^k + x^(2 k) + x^(3 k) + 2 x^(4 k))/(k (1 - x^(5 k))), {k, 1, nmax}]], {x, 0, nmax}], x]
Table[Sum[PartitionsP[k] PartitionsP[n - 5 k], {k, 0, n/5}], {n, 0, 48}]

Crossrefs

Cf. A000712, A002513, A030202, A030205, A145466, A318026, A318027.

Sequence in context: A266773 A347448 A024789 * A373296 A200661 A175539

Adjacent sequences: A318025 A318026 A318027 * A318029 A318030 A318031

Keyword

nonn

Author

Ilya Gutkovskiy, Aug 13 2018

Status

approved