A109701 Number of partitions of n into parts each equal to 1 mod 6.

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 4, 5, 6, 7, 7, 7, 7, 8, 10, 11, 12, 12, 12, 13, 15, 17, 18, 19, 19, 20, 23, 26, 28, 29, 30, 31, 34, 38, 41, 43, 44, 46, 50, 55, 60, 63, 65, 67, 72, 79, 85, 90, 93, 96, 102, 111, 120, 127, 132, 136, 143, 154, 166, 176, 183, 189, 198

Offset

0,8

Comments

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

Links

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

Formula

G.f.: 1/Product_{j >= 0} (1-x^(1+6j)). - Emeric Deutsch, Apr 14 2006

a(n) ~ Gamma(1/6) * exp(Pi*sqrt(n)/3) / (4 * sqrt(3) * Pi^(5/6) * n^(7/12)) * (1 - (7/(24*Pi) + Pi/144) / sqrt(n)). - Vaclav Kotesovec, Feb 27 2015, extended Jan 24 2017

a(n) = (1/n)*Sum_{k=1..n} A284098(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^(6*j)). - Ilya Gutkovskiy, Jul 17 2019

Example

a(10)=2 since 10 = 7+1+1+1 = 1+1+1+1+1+1+1+1+1+1

Maple

g:=1/product(1-x^(1+6*j), j=0..20): gser:=series(g, x=0, 77): seq(coeff(gser, x, n), n=0..74); # Emeric Deutsch, Apr 14 2006

Mathematica

nmax=100; CoefficientList[Series[Product[1/(1-x^(6*k+1)), {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 27 2015 *)

Crossrefs

Cf. A284098.

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), A109697 (m=5), this sequence (m=6), A109703 (m=7), A277090 (m=8).

Sequence in context: A374032 A317245 A332277 * A124751 A103374 A208251

Adjacent sequences: A109698 A109699 A109700 * A109702 A109703 A109704

Keyword

nonn

Author

Erich Friedman, Aug 07 2005

Extensions

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

Status

approved