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%I #41 May 08 2022 08:42:42
%S 1,0,1,0,2,1,0,5,3,1,0,15,10,6,1,0,52,45,25,10,1,0,203,241,100,65,15,
%T 1,0,877,1428,511,350,140,21,1,0,4140,9325,3626,1736,1050,266,28,1,0,
%U 21147,67035,29765,9030,6951,2646,462,36,1,0,115975,524926,250200,60355,42651,22827,5880,750,45,1
%N Number T(n,k) of set partitions of [n] where each subset is again partitioned into k nonempty subsets; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%H Alois P. Heinz, <a href="/A324162/b324162.txt">Rows n = 0..140, flattened</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F E.g.f. of column k>0: exp((exp(x)-1)^k/k!).
%F Sum_{k=1..n} k * T(n,k) = A325929(n).
%F T(n,k) = Sum_{j=0..floor(n/k)} (k*j)! * Stirling2(n,k*j)/(k!^j * j!) for k > 0. - _Seiichi Manyama_, May 07 2022
%e T(4,2) = 10: 123/4, 124/3, 12/34, 134/2, 13/24, 14/23, 1/234, 1/2|3/4, 1/3|2/4, 1/4|2/3.
%e Triangle T(n,k) begins:
%e 1;
%e 0, 1;
%e 0, 2, 1;
%e 0, 5, 3, 1;
%e 0, 15, 10, 6, 1;
%e 0, 52, 45, 25, 10, 1;
%e 0, 203, 241, 100, 65, 15, 1;
%e 0, 877, 1428, 511, 350, 140, 21, 1;
%e 0, 4140, 9325, 3626, 1736, 1050, 266, 28, 1;
%e ...
%p T:= proc(n, k) option remember; `if`(n=0, 1, `if`(k=0, 0, add(
%p T(n-j, k)*binomial(n-1, j-1)*Stirling2(j, k), j=k..n)))
%p end:
%p seq(seq(T(n, k), k=0..n), n=0..12);
%t nmax = 10;
%t col[k_] := col[k] = CoefficientList[Exp[(Exp[x]-1)^k/k!] + O[x]^(nmax+1), x][[k+1;;]] Range[k, nmax]!;
%t T[n_, k_] := Which[k == n, 1, k == 0, 0, True, col[k][[n-k+1]]];
%t Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Apr 26 2020 *)
%o (PARI) T(n, k) = if(k==0, 0^n, sum(j=0, n\k, (k*j)!*stirling(n, k*j, 2)/(k!^j*j!))); \\ _Seiichi Manyama_, May 07 2022
%Y Columns k=0-10 give: A000007, A000110 (for n>0), A060311, A327504, A327505, A327506, A327507, A327508, A327509, A327510, A327511.
%Y Row sums give A324238.
%Y T(2n,n) gives A324241.
%Y Cf. A008277, A048993, A039810, A130191, A325929.
%K nonn,tabl
%O 0,5
%A _Alois P. Heinz_, Sep 02 2019
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