A275429 Number of set partitions of [n] such that n is a multiple of each block size.

1, 1, 2, 2, 11, 2, 167, 2, 1500, 1206, 16175, 2, 3486584, 2, 3188421, 29226654, 772458367, 2, 130880325103, 2, 4173951684174, 623240762412, 644066092301, 2, 220076136813712815, 31580724696908, 538897996103277, 49207275464475052, 44147498142028751570, 2

Offset: 0

Author

Alois P. Heinz, Jul 27 2016

Keywords

• nonn

Examples

			a(4) = 11: 1234, 12|34, 12|3|4, 13|24, 13|2|4, 14|23, 1|23|4, 14|2|3, 1|24|3, 1|2|34, 1|2|3|4.
a(5) = 2: 12345, 1|2|3|4|5.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..587
• Wikipedia, Partition of a set

Crossrefs

Main diagonal of A275422.

Programs

Maple

A:= proc(n, k) option remember; `if`(n=0, 1, add(
      `if`(j>n, 0, A(n-j, k)*binomial(n-1, j-1)), j=
      `if`(k=0, 1..n, numtheory[divisors](k))))
    end:
a:= n-> A(n$2):
seq(a(n), n=0..30);

Mathematica

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

Formula

a(n) = n! * [x^n] exp(Sum_{d|n} x^d/d!) for n>0, a(0) = 1.

a(n) = A275422(n,n).

a(p) = 2 for p prime.