A219601 Number of partitions of n in which no parts are multiples of 6.
1, 1, 2, 3, 5, 7, 10, 14, 20, 27, 37, 49, 65, 85, 111, 143, 184, 234, 297, 374, 470, 586, 729, 902, 1113, 1367, 1674, 2042, 2485, 3013, 3645, 4395, 5288, 6344, 7595, 9070, 10809, 12852, 15252, 18062, 21352, 25191, 29671, 34884, 40948, 47985, 56146, 65592
Offset: 0
Examples
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.
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
Crossrefs
Programs
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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 *) -
PARI
for(n=0, 47, A=x*O(x^n); print1(polcoeff(eta(x^6+A)/eta(x+A), n), ", "))
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
Comments