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

0,3

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

Cf. A097797.

Cf. A261770, A261736, A320608.

Number of r-regular partitions for r = 2 through 12: A000009, A000726, A001935, A035959, A219601, A035985, A261775, A104502, A261776, A328545, A328546.

Sequence in context: A326977 A035967 A097797 * A035975 A035984 A035994

Adjacent sequences: A219598 A219599 A219600 * A219602 A219603 A219604

Keyword

easy,nonn

Author

Arkadiusz Wesolowski, Nov 23 2012

Status

approved