A275044 Number of set partitions of [n^2] such that within each block the numbers of elements from all residue classes modulo n are equal for n>0, a(0)=1.

1, 1, 3, 64, 25097, 350813126, 293327384637282, 22208366234650578141209, 213426677887357366350726096998529, 344735749788852590196707169431958672823413322, 118966637603805785518622376062965559343297730169187276656138

Offset: 0

Author

Alois P. Heinz, Jul 14 2016

Keywords

• nonn

Examples

			a(2) = 3: 1234, 12|34, 14|23.
a(3) = 64: 123456789, 123456|789, 123459|678, 123468|579, ... , 159|267|348, 168|279|345, 189|267|345.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..30
• Wikipedia, Partition of a set

Crossrefs

Main diagonal of A275043.

Programs

Maple

b:= proc(n, k) option remember; `if`(k*n=0, 1, add(
       binomial(n, j)^k*(n-j)*b(j, k), j=0..n-1)/n)
    end:
a:= n-> b(n$2):
seq(a(n), n=0..12);

Mathematica

b[n_, k_] := b[n, k] = If[k*n == 0, 1, Sum[Binomial[n, j]^k*(n-j)*b[j, k], {j, 0, n-1}]/n];
a[n_] := b[n, n];
Table[a[n], {n, 0, 12}] (* Jean-François Alcover, May 27 2018, translated from Maple *)

Formula

a(n) = (n!)^n * [x^n] exp(Sum_{k>=1} x^k / (k!)^n). - Ilya Gutkovskiy, Jul 12 2020