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

0,6

Links

Alois P. Heinz, Table of n, a(n) for n = 0..619

Wikipedia, Partition of a set

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.

Example

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

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 *)

Prog

(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

Crossrefs

Column k=5 of A275422.

Sequence in context: A365736 A364522 A226910 * A099131 A212384 A306347

Adjacent sequences: A275420 A275421 A275422 * A275424 A275425 A275426

Keyword

nonn

Author

Alois P. Heinz, Jul 27 2016

Status

approved