cp's OEIS Frontend

A219601 Number of partitions of n in which no parts are multiples of 6.

%I A219601 #35 Feb 16 2025 08:33:18
%S A219601 1,1,2,3,5,7,10,14,20,27,37,49,65,85,111,143,184,234,297,374,470,586,
%T A219601 729,902,1113,1367,1674,2042,2485,3013,3645,4395,5288,6344,7595,9070,
%U A219601 10809,12852,15252,18062,21352,25191,29671,34884,40948,47985,56146,65592
%N A219601 Number of partitions of n in which no parts are multiples of 6.
%C A219601 Also partitions where parts are repeated at most 5 times. [_Joerg Arndt_, Dec 31 2012]
%H A219601 Seiichi Manyama, <a href="/A219601/b219601.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Arkadiusz Wesolowski)
%H A219601 Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 15.
%H A219601 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PartitionFunctionb.html">Partition Function b_k</a>
%F A219601 G.f.: P(x^6)/P(x), where P(x) = prod(k>=1, 1-x^k).
%F A219601 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
%F A219601 a(n) = (1/n)*Sum_{k=1..n} A284326(k)*a(n-k), a(0) = 1. - _Seiichi Manyama_, Mar 25 2017
%e A219601 7 = 7
%e A219601   = 5 + 2
%e A219601   = 5 + 1 + 1
%e A219601   = 4 + 3
%e A219601   = 4 + 2 + 1
%e A219601   = 4 + 1 + 1 + 1
%e A219601   = 3 + 3 + 1
%e A219601   = 3 + 2 + 2
%e A219601   = 3 + 2 + 1 + 1
%e A219601   = 3 + 1 + 1 + 1 + 1
%e A219601   = 2 + 2 + 2 + 1
%e A219601   = 2 + 2 + 1 + 1 + 1
%e A219601   = 2 + 1 + 1 + 1 + 1 + 1
%e A219601   = 1 + 1 + 1 + 1 + 1 + 1 + 1
%e A219601 so a(7) = 14.
%t A219601 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 *)
%t A219601 Table[Count[IntegerPartitions@n, x_ /; ! MemberQ [Mod[x, 6], 0, 2] ], {n, 0, 47}] (* _Robert Price_, Jul 28 2020 *)
%o A219601 (PARI) for(n=0, 47, A=x*O(x^n); print1(polcoeff(eta(x^6+A)/eta(x+A), n), ", "))
%Y A219601 Cf. A097797.
%Y A219601 Cf. A261770, A261736, A320608.
%Y A219601 Number of r-regular partitions for r = 2 through 12: A000009, A000726, A001935, A035959, A219601, A035985, A261775, A104502, A261776, A328545, A328546.
%K A219601 easy,nonn
%O A219601 0,3
%A A219601 _Arkadiusz Wesolowski_, Nov 23 2012