A109697 Number of partitions of n into parts each equal to 1 mod 5.

1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 5, 6, 7, 7, 7, 8, 10, 11, 12, 12, 13, 15, 17, 18, 19, 20, 23, 26, 28, 29, 31, 34, 38, 41, 43, 45, 50, 55, 60, 63, 66, 71, 79, 85, 90, 94, 101, 110, 120, 127, 133, 141, 153, 165, 176, 184, 195, 210, 227, 241, 254, 267, 286, 307, 327

Offset: 0

Author

Erich Friedman, Aug 07 2005

Keywords

• nonn

Examples

			a(11)=3 since 11 = 11 = 6+1+1+1+1+1 = 1+1+1+1+1+1+1+1+1+1+1
		

Links

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

Crossrefs

Cf. A000041, A003105, A284097.

Cf. similar sequences of number of partitions of n into parts congruent to 1 mod m: A000009 (m=2), A035382 (m=3), A035451 (m=4), this sequence (m=5), A109701 (m=6), A109703 (m=7), A277090 (m=8).

Programs

Maple

g:=1/product(1-x^(1+5*j),j=0..25): gser:=series(g,x=0,85): seq(coeff(gser,x,n),n=0..80); # Emeric Deutsch, Mar 30 2006

Mathematica

Table[Count[IntegerPartitions[n],?(Union[Mod[#,5]]=={1}&)],{n,0,75}] (* _Harvey P. Dale, Oct 08 2011 *)

Formula

G.f.: 1/product(1-x^(1+5j), j=0..infinity). - Emeric Deutsch, Mar 30 2006

a(n) ~ Gamma(1/5) * exp(Pi*sqrt(2*n/15)) / (2^(8/5) * 3^(1/10) * 5^(2/5) * Pi^(4/5) * n^(3/5)) * (1 - (3*sqrt(3/10)/(5*Pi) + Pi/(120*sqrt(30))) / sqrt(n)). - Vaclav Kotesovec, Feb 27 2015, extended Jan 24 2017

a(n) = (1/n)*Sum_{k=1..n} A284097(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 20 2017

G.f.: Sum_{k>=0} x^k / Product_{j=1..k} (1 - x^(5*j)). - Ilya Gutkovskiy, Jul 17 2019

Extensions

More terms from Emeric Deutsch, Mar 30 2006