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 A229243 Number A(n,k) of set partitions of {1,...,k*n} into sets of size at most n; square array A(n,k), n>=0, k>=0, read by antidiagonals. 6
 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 10, 5, 1, 1, 1, 76, 166, 15, 1, 1, 1, 764, 12644, 3795, 52, 1, 1, 1, 9496, 1680592, 3305017, 112124, 203, 1, 1, 1, 140152, 341185496, 6631556521, 1245131903, 4163743, 877, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,9 LINKS Alois P. Heinz, Antidiagonals n = 0..30, flattened FORMULA A(n,k) = (n*k)! * [x^(n*k)] exp(Sum_{j=1..n} x^j/j!). A(n,k) = A229223(n*k,n). EXAMPLE Square array A(n,k) begins: 1, 1, 1, 1, 1, 1, ... 1, 1, 1, 1, 1, 1, ... 1, 2, 10, 76, 764, 9496, ... 1, 5, 166, 12644, 1680592, 341185496, ... 1, 15, 3795, 3305017, 6631556521, 25120541332271, ... 1, 52, 112124, 1245131903, 41916097982471, 3282701194678476257, ... MAPLE G:= proc(n, k) option remember; local j; if k>n then G(n, n) elif n=0 then 1 elif k<1 then 0 else G(n-k, k); for j from k-1 to 1 by -1 do %*(n-j)/j +G(n-j, k) od; % fi end: A:= (n, k)-> G(n*k, n): seq(seq(A(n, d-n), n=0..d), d=0..10); MATHEMATICA G[n_, k_] := G[n, k] = Module[{j, g}, Which[k > n, G[n, n], n == 0, 1, k < 1, 0, True, g = G[n-k, k]; For[j = k-1, j >= 1, j--, g = g*(n-j)/j + G[n-j, k] ]; g ] ]; A[n_, k_] := G[n*k, n]; Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Dec 23 2013, translated from Maple *) CROSSREFS Columns k=0-3 give: A000012, A000110, A229228, A229413. Rows n=0+1, 2-3 give: A000012, A066223, A229414. Main diagonal gives: A229229. Cf. A229223. Sequence in context: A156188 A179930 A007737 * A105688 A066017 A236938 Adjacent sequences: A229240 A229241 A229242 * A229244 A229245 A229246 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Sep 17 2013 STATUS approved

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Last modified October 1 11:17 EDT 2023. Contains 365826 sequences. (Running on oeis4.)