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

0,17

Links

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

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

Example

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

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 *)

Prog

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

Crossrefs

Cf. A284281.

Sequence in context: A097368 A271900 A194083 * A301563 A330896 A328294

Adjacent sequences: A109696 A109697 A109698 * A109700 A109701 A109702

Keyword

nonn

Author

Erich Friedman, Aug 07 2005

Extensions

More terms from Emeric Deutsch, Mar 30 2006

Status

approved