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A324162
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.
16
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, 1, 0, 877, 1428, 511, 350, 140, 21, 1, 0, 4140, 9325, 3626, 1736, 1050, 266, 28, 1, 0, 21147, 67035, 29765, 9030, 6951, 2646, 462, 36, 1, 0, 115975, 524926, 250200, 60355, 42651, 22827, 5880, 750, 45, 1
OFFSET
0,5
LINKS
FORMULA
E.g.f. of column k>0: exp((exp(x)-1)^k/k!).
Sum_{k=1..n} k * T(n,k) = A325929(n).
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
EXAMPLE
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.
Triangle T(n,k) begins:
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, 1;
0, 877, 1428, 511, 350, 140, 21, 1;
0, 4140, 9325, 3626, 1736, 1050, 266, 28, 1;
...
MAPLE
T:= proc(n, k) option remember; `if`(n=0, 1, `if`(k=0, 0, add(
T(n-j, k)*binomial(n-1, j-1)*Stirling2(j, k), j=k..n)))
end:
seq(seq(T(n, k), k=0..n), n=0..12);
MATHEMATICA
nmax = 10;
col[k_] := col[k] = CoefficientList[Exp[(Exp[x]-1)^k/k!] + O[x]^(nmax+1), x][[k+1;; ]] Range[k, nmax]!;
T[n_, k_] := Which[k == n, 1, k == 0, 0, True, col[k][[n-k+1]]];
Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 26 2020 *)
PROG
(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
CROSSREFS
Columns k=0-10 give: A000007, A000110 (for n>0), A060311, A327504, A327505, A327506, A327507, A327508, A327509, A327510, A327511.
Row sums give A324238.
T(2n,n) gives A324241.
Sequence in context: A083417 A021479 A073583 * A060136 A327358 A256664
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Sep 02 2019
STATUS
approved