A261776 Expansion of Product_{k>=1} (1 - x^(10*k))/(1 - x^k).

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 41, 55, 75, 98, 130, 169, 220, 282, 363, 460, 584, 735, 923, 1151, 1435, 1775, 2194, 2698, 3311, 4045, 4935, 5994, 7270, 8787, 10600, 12749, 15310, 18330, 21912, 26130, 31107, 36949, 43823, 51863, 61290, 72293, 85145, 100107

Offset: 0

Author

Vaclav Kotesovec, Aug 31 2015

Keywords

• nonn

Comments

General asymptotic formula (Hagis, 1971): If s > 1 and g.f. = Product_{k>=1} (1 - x^(s*k))/(1 - x^k), then a(n) ~ exp(Pi*sqrt(2*n*(s-1)/(3*s))) * (s-1)^(1/4) / (2 * 6^(1/4) * s^(3/4) * n^(3/4)) * (1 + ((s-1)^(3/2)*Pi/(24*sqrt(6*s)) - 3*sqrt(6*s) / (16*Pi * sqrt(s-1))) / sqrt(n) + ((s-1)^3*Pi^2/(6912*s) - 45*s/(256*(s-1)*Pi^2) - 5*(s-1)/128) / n), minor asymptotic terms added by Vaclav Kotesovec, Jan 13 2017

The formula in the article by Noureddine Chair: "The Euler-Riemann Gases, and Partition Identities", p. 32, is incorrect (must be s -> s-1 and 24 -> 24*n).

Number of partitions in which no part occurs more than 9 times. - Ilya Gutkovskiy, May 31 2017

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• Noureddine Chair, The Euler-Riemann Gases, and Partition Identities, arXiv:1306.5415 [math-ph], 2013, p. 32.
• Peter Hagis jr., Partitions with a restriction on the multiplicity of the summands, Transactions of the American Mathematical Society, Volume 155, Number 2, April 1971.
• Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 15.

Crossrefs

Number of r-regular partitions for r = 2 through 12: A000009, A000726, A001935, A035959, A219601, A035985, A261775, A104502, A261776, A328545, A328546.

Cf. A261772, A145707, A320612.

Programs

Mathematica

nmax = 50; CoefficientList[Series[Product[(1 - x^(10*k))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Table[Count[IntegerPartitions@n, x_ /; ! MemberQ [Mod[x, 10], 0, 2] ], {n, 0, 47}] (* Robert Price, Jul 29 2020 *)

PARI

Vec(prod(k=1, 51, (1 - x^(10*k))/(1 - x^k)) + O(x^51)) \\ Indranil Ghosh, Mar 25 2017

Formula

a(n) ~ 3*Pi * BesselI(1, sqrt((24*n + 9)/10) * Pi/2) / (5*sqrt(24*n + 9)) ~ exp(Pi*sqrt(3*n/5)) * 3^(1/4) / (4 * 5^(3/4) * n^(3/4)) * (1 + (3^(3/2)*Pi/(16*sqrt(5)) - sqrt(15)/(8*Pi)) / sqrt(n) + (27*Pi^2/2560 - 25/(128*Pi^2) - 45/128) / n). - Vaclav Kotesovec, Aug 31 2015, extended Jan 14 2017

a(n) = (1/n)*Sum_{k=1..n} A284344(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017