OFFSET
0,3
COMMENTS
Also partitions where parts are repeated at most 5 times. [Joerg Arndt, Dec 31 2012]
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Arkadiusz Wesolowski)
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 15.
Eric Weisstein's World of Mathematics, Partition Function b_k
FORMULA
G.f.: P(x^6)/P(x), where P(x) = prod(k>=1, 1-x^k).
a(n) ~ Pi*sqrt(5) * BesselI(1, sqrt(5*(24*n + 5)/6) * Pi/6) / (3*sqrt(24*n + 5)) ~ exp(Pi*sqrt(5*n)/3) * 5^(1/4) / (12 * n^(3/4)) * (1 + (5^(3/2)*Pi/144 - 9/(8*Pi*sqrt(5))) / sqrt(n) + (125*Pi^2/41472 - 27/(128*Pi^2) - 25/128) / n). - Vaclav Kotesovec, Aug 31 2015, extended Jan 14 2017
a(n) = (1/n)*Sum_{k=1..n} A284326(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017
EXAMPLE
7 = 7
= 5 + 2
= 5 + 1 + 1
= 4 + 3
= 4 + 2 + 1
= 4 + 1 + 1 + 1
= 3 + 3 + 1
= 3 + 2 + 2
= 3 + 2 + 1 + 1
= 3 + 1 + 1 + 1 + 1
= 2 + 2 + 2 + 1
= 2 + 2 + 1 + 1 + 1
= 2 + 1 + 1 + 1 + 1 + 1
= 1 + 1 + 1 + 1 + 1 + 1 + 1
so a(7) = 14.
MATHEMATICA
m = 47; f[x_] := (x^6 - 1)/(x - 1); g[x_] := Product[f[x^k], {k, 1, m}]; CoefficientList[Series[g[x], {x, 0, m}], x] (* Arkadiusz Wesolowski, Nov 27 2012 *)
Table[Count[IntegerPartitions@n, x_ /; ! MemberQ [Mod[x, 6], 0, 2] ], {n, 0, 47}] (* Robert Price, Jul 28 2020 *)
PROG
(PARI) for(n=0, 47, A=x*O(x^n); print1(polcoeff(eta(x^6+A)/eta(x+A), n), ", "))
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Arkadiusz Wesolowski, Nov 23 2012
STATUS
approved