A275043 Number A(n,k) of set partitions of [k*n] such that within each block the numbers of elements from all residue classes modulo k are equal for k>0, A(n,0)=1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 5, 1, 1, 1, 5, 16, 15, 1, 1, 1, 9, 64, 131, 52, 1, 1, 1, 17, 298, 1613, 1496, 203, 1, 1, 1, 33, 1540, 25097, 69026, 22482, 877, 1, 1, 1, 65, 8506, 461105, 4383626, 4566992, 426833, 4140, 1, 1, 1, 129, 48844, 9483041, 350813126, 1394519922, 437665649, 9934563, 21147, 1

Offset

0,9

Links

Alois P. Heinz, Antidiagonals n = 0..60, flattened

J.-M. Sixdeniers, K. A. Penson and A. I. Solomon, Extended Bell and Stirling Numbers From Hypergeometric Exponentiation, J. Integer Seqs. Vol. 4 (2001), #01.1.4.

Wikipedia, Partition of a set

Example

A(2,2) = 3: 1234, 12|34, 14|23.
A(2,3) = 5: 123456, 123|456, 126|345, 135|246, 156|234.
A(2,4) = 9: 12345678, 1234|5678, 1238|4567, 1247|3568, 1278|3456, 1346|2578, 1368|2457, 1467|2358, 1678|2345.
A(3,2) = 16: 123456, 1234|56, 1236|45, 1245|36, 1256|34, 12|3456, 12|34|56, 12|36|45, 1346|25, 1456|23, 14|2356, 14|23|56, 16|2345, 16|23|45, 14|25|36, 16|25|34.
Square array A(n,k) begins:
  1,   1,     1,       1,          1,            1,               1, ...
  1,   1,     1,       1,          1,            1,               1, ...
  1,   2,     3,       5,          9,           17,              33, ...
  1,   5,    16,      64,        298,         1540,            8506, ...
  1,  15,   131,    1613,      25097,       461105,         9483041, ...
  1,  52,  1496,   69026,    4383626,    350813126,     33056715626, ...
  1, 203, 22482, 4566992, 1394519922, 573843627152, 293327384637282, ...

Maple

A:= proc(n, k) option remember; `if`(k*n=0, 1, add(
       binomial(n, j)^k*(n-j)*A(j, k), j=0..n-1)/n)
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);

Mathematica

A[n_, k_] := A[n, k] = If[k*n == 0, 1, Sum[Binomial[n, j]^k*(n-j)*A[j, k], {j, 0, n-1}]/n]; Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jan 17 2017, translated from Maple *)

Crossrefs

Columns k=0-10 give: A000012, A000110, A023998, A061684, A061685, A061686, A061687, A061688, A275097, A275098, A275099.

Rows n=0+1,2-5 give: A000012, A094373, A275100, A275101, A275102.

Main diagonal gives A275044.

Cf. A345400.

Sequence in context: A368025 A290759 A306245 * A227061 A201949 A291709

Adjacent sequences: A275040 A275041 A275042 * A275044 A275045 A275046

Keyword

nonn,tabl

Author

Alois P. Heinz, Jul 14 2016

Status

approved