A109699 Number of partitions of n into parts each equal to 3 mod 5.

1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 3, 2, 2, 4, 2, 4, 4, 3, 5, 4, 5, 6, 5, 7, 6, 8, 8, 7, 11, 9, 10, 13, 10, 14, 14, 14, 17, 16, 19, 19, 20, 24, 21, 27, 27, 27, 33, 30, 35, 38, 36, 44, 42, 47, 51, 50, 58, 57, 63, 68, 66, 79, 76, 82, 92, 88, 101, 104, 107, 120

Offset: 0

Author

Erich Friedman, Aug 07 2005

Keywords

• nonn

Examples

			a(21)=3 since 21 = 18+3 = 13+8 = 3+3+3+3+3+3+3
		

Links

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

Crossrefs

Cf. A284281.

Programs

Maple

g:=1/product(1-x^(3+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

nmax=100; CoefficientList[Series[Product[1/(1-x^(5*k+3)),{k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 27 2015 *)
Join[{1},Table[Length[Select[IntegerPartitions[n],Union[Mod[#,5]]=={3}&]],{n,80}]] (* Harvey P. Dale, Dec 01 2024 *)

PARI

Vec(prod(k=0, 100, 1/(1 - x^(5*k + 3))) + O(x^111)) \\ Indranil Ghosh, Mar 24 2017

Formula

G.f.: 1/product(1-x^(3+5j), j=0..infinity). - Emeric Deutsch, Mar 30 2006

a(n) ~ Gamma(3/5) * exp(Pi*sqrt(2*n/15)) / (2^(9/5) * 3^(3/10) * 5^(1/5) * Pi^(2/5) * n^(4/5)) * (1 + (11*Pi/(120*sqrt(30)) - 6*sqrt(6/5)/(5*Pi)) / sqrt(n)). - Vaclav Kotesovec, Feb 27 2015, extended Jan 24 2017

a(n) = (1/n)*Sum_{k=1..n} A284281(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 24 2017

Extensions

More terms from Emeric Deutsch, Mar 30 2006