A275423 Number of set partitions of [n] such that five is a multiple of each block size.

1, 1, 1, 1, 1, 2, 7, 22, 57, 127, 379, 1849, 9109, 37324, 128129, 507508, 3031393, 19609773, 108440893, 500515633, 2467616641, 17154715726, 134519207131, 927764339426, 5359830269641, 31580724696907, 248587878630807, 2259650025239257, 18541914182165557

Offset: 0

Author

Alois P. Heinz, Jul 27 2016

Keywords

• nonn

Examples

			a(6) = 7: 12345|6, 12346|5, 12356|4, 12456|3, 13456|2, 1|23456, 1|2|3|4|5|6.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..619
• Wikipedia, Partition of a set

Crossrefs

Column k=5 of A275422.

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 1, add(
      `if`(j>n, 0, a(n-j)*binomial(n-1, j-1)), j=[1, 5]))
    end:
seq(a(n), n=0..30);
# second Maple program:
seq(simplify(hypergeom([-n/5, (1-n)/5, (2-n)/5, (3-n)/5, (4-n)/5], [], -625/24)), n = 0..28); # Karol A. Penson, Sep 14 2023.

Mathematica

a[n_] := a[n] = If[n == 0, 1, Sum[If[j > n, 0, a[n-j]*Binomial[n-1, j-1]], {j, {1, 5}}]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 17 2018, translated from Maple *)

PARI

a(n) = n!*sum(k=0, n\5, 1/5!^k*binomial(n-4*k, k)/(n-4*k)!); \\ Seiichi Manyama, Feb 26 2022

PARI

a(n) = if(n<5, 1, a(n-1)+binomial(n-1, 4)*a(n-5)); \\ Seiichi Manyama, Feb 26 2022

Formula

E.g.f.: exp(x+x^5/5!).

From Seiichi Manyama, Feb 26 2022: (Start)

a(n) = n! * Sum_{k=0..floor(n/5)} (1/5!)^k * binomial(n-4*k,k)/(n-4*k)!.

a(n) = a(n-1) + binomial(n-1,4) * a(n-5) for n > 4. (End)

a(n) = hypergeom([-n/5,(1-n)/5,(2-n)/5,(3-n)/5,(4-n)/5],[],-625/24). - Karol A. Penson, Sep 14 2023.