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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. 15
 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-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: A124560 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

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Last modified July 23 12:19 EDT 2021. Contains 346259 sequences. (Running on oeis4.)