A320608 -id:A320608 - cp's OEIS frontend

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

Arkadiusz Wesolowski, Nov 23 2012

Also partitions where parts are repeated at most 5 times. [Joerg Arndt, Dec 31 2012]

			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.

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

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.

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 *)

for(n=0, 47, A=x*O(x^n); print1(polcoeff(eta(x^6+A)/eta(x+A), n), ", "))

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

A210485 Number T(n,k) of parts in all partitions of n in which no part occurs more than k times; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

0, 0, 1, 0, 1, 3, 0, 3, 3, 6, 0, 3, 8, 8, 12, 0, 5, 11, 15, 15, 20, 0, 8, 17, 24, 29, 29, 35, 0, 10, 23, 36, 41, 47, 47, 54, 0, 13, 36, 50, 65, 71, 78, 78, 86, 0, 18, 48, 75, 91, 104, 111, 119, 119, 128, 0, 25, 69, 102, 132, 150, 165, 173, 182, 182, 192

Offset: 0

Alois P. Heinz, Jan 23 2013

T(n,k) is defined for n,k >= 0. The triangle contains terms with k <= n. T(n,k) = T(n,n) = A006128(n) for k >= n.

For fixed k > 0, T(n,k) ~ 3^(1/4) * log(k+1) * exp(Pi*sqrt(2*k*n/(3*(k+1)))) / (Pi * (8*k*(k+1)*n)^(1/4)). - Vaclav Kotesovec, Oct 18 2018

			T(6,2) = 17: [6], [5,1], [4,2], [3,3], [4,1,1], [3,2,1], [2,2,1,1].
Triangle T(n,k) begins:
  0;
  0,  1;
  0,  1,  3;
  0,  3,  3,  6;
  0,  3,  8,  8, 12;
  0,  5, 11, 15, 15, 20;
  0,  8, 17, 24, 29, 29, 35;
  0, 10, 23, 36, 41, 47, 47, 54;
  0, 13, 36, 50, 65, 71, 78, 78, 86;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0-10 give: A000004, A015723, A185350, A117148, A320607, A320608, A320609, A320610, A320611, A320612, A320613.
Main diagonal gives A006128.
T(2n,n) gives A364245.
Cf. A213177, A286653.

b:= proc(n, i, k) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->[l[1], l[2]+l[1]*j])(b(n-i*j, i-1, k)), j=0..min(n/i, k))))
    end:
T:= (n, k)-> b(n, n, k)[2]:
seq(seq(T(n, k), k=0..n), n=0..12);

b[n_, i_, k_] := b[n, i, k] = If[n == 0, {1, 0}, If[i < 1, {0, 0}, Sum[b[n-i*j, i-1, k] /. l_List :> {l[[1]], l[[2]] + l[[1]]*j}, {j, 0, Min[n/i, k]}]]]; T[n_, k_] := b[n, n, k][[2]]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 27 2013, translated from Maple *)

T(n,k) = Sum_{i=0..k} A213177(n,i).

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula