A109702 Number of partitions of n into parts each equal to 5 mod 6.

1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 2, 1, 0, 1, 1, 2, 2, 1, 1, 1, 2, 3, 3, 2, 1, 2, 3, 4, 4, 2, 2, 3, 5, 6, 5, 3, 3, 5, 7, 8, 6, 4, 5, 8, 10, 10, 8, 6, 8, 11, 13, 13, 10, 9, 12, 15, 18, 17, 14, 13, 16, 21, 23, 22, 18, 18, 23, 28, 31, 28, 24, 25, 31, 38, 39, 36, 32, 34

Offset: 0

Author

Erich Friedman, Aug 07 2005

Keywords

• nonn

Examples

			a(40)=4 since 40 = 35+5 = 29+11 = 23+17 = 5+5+5+5+5+5+5+5.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A284104.

Cf. similar sequences of number of partitions of n into parts congruent to m-1 mod m: A000009 (m=2), A035386 (m=3), A035462 (m=4), A109700 (m=5), this sequence (m=6), A109708 (m=7).

Programs

Maple

g:=1/product(1-x^(5+6*j),j=0..20): gser:=series(g,x=0,92): seq(coeff(gser,x,n),n=0..89); # Emeric Deutsch, Apr 14 2006

Mathematica

nmax=100; CoefficientList[Series[Product[1/(1-x^(6*k+5)),{k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 27 2015 *)
Table[Count[IntegerPartitions[n],?(Union[Mod[#,6]]=={5}&)],{n,0,90}] (* _Harvey P. Dale, Mar 08 2022 *)

Formula

G.f.: 1/product(1-x^(5+6j),j=0..infinity). - Emeric Deutsch, Apr 14 2006

a(n) ~ Gamma(5/6) * exp(Pi*sqrt(n)/3) / (4 * sqrt(3) * Pi^(1/6) * n^(11/12)) * (1 - (55/(24*Pi) + Pi/144) / sqrt(n)). - Vaclav Kotesovec, Feb 27 2015, extended Jan 24 2017

a(n) = (1/n)*Sum_{k=1..n} A284104(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 20 2017

Euler transform of period 6 sequence [ 0, 0, 0, 0, 1, 0, ...]. - Kevin T. Acres, Apr 28 2018

Extensions

Changed offset to 0 and added a(0)=1 by Vaclav Kotesovec, Feb 27 2015