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

0,7

Links

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

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

Example

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

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 *)

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).

Sequence in context: A025783 A025780 A199121 * A358903 A103373 A038539

Adjacent sequences: A109694 A109695 A109696 * A109698 A109699 A109700

Keyword

nonn

Author

Erich Friedman, Aug 07 2005

Extensions

More terms from Emeric Deutsch, Mar 30 2006

Status

approved