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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. 14
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 (list; table; graph; refs; listen; history; text; internal format)
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-Fran├žois 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.

Sequence in context: A144150 A124560 A290759 * A227061 A201949 A291709

Adjacent sequences:  A275040 A275041 A275042 * A275044 A275045 A275046

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Jul 14 2016

STATUS

approved

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Last modified March 22 12:46 EDT 2019. Contains 321421 sequences. (Running on oeis4.)