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A330759 Number T(n,k) of set partitions into k blocks of strict integer partitions of n; triangle T(n,k), n>=0, 0<=k<=A003056(n), read by rows.

%I A330759 #33 Mar 12 2021 14:07:01
%S A330759 1,0,1,0,1,0,2,1,0,2,1,0,3,2,0,4,5,1,0,5,6,1,0,6,9,2,0,8,13,3,0,10,23,
%T A330759 10,1,0,12,27,11,1,0,15,40,19,2,0,18,51,26,3,0,22,71,40,5,0,27,100,73,
%U A330759 16,1,0,32,127,93,19,1,0,38,163,132,31,2,0,46,215,184,45,3
%N A330759 Number T(n,k) of set partitions into k blocks of strict integer partitions of n; triangle T(n,k), n>=0, 0<=k<=A003056(n), read by rows.
%H A330759 Alois P. Heinz, <a href="/A330759/b330759.txt">Rows n = 0..1000, flattened</a>
%H A330759 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A330759 Sum_{k=0..2} T(n,k) = A072706(n).
%F A330759 Sum_{k=1..A003056(n)} k * T(n,k) = A330765(n).
%F A330759 T(A000217(n),n) = 1.
%F A330759 T(A000096(n),n) = A000041(n).
%F A330759 T(n*(n+1)/2+j,n) = A000041(j) for 0 <= j <= n.
%e A330759 T(10,1) = 10: (10), 1234, 127, 136, 145, 19, 235, 28, 37, 46.
%e A330759 T(10,2) = 23: 123|4, 124|3, 12|34, 12|7, 134|2, 13|24, 13|6, 14|23, 14|5, 15|4, 16|3, 17|2, 1|234, 1|27, 1|36, 1|45, 1|9, 23|5, 25|3, 2|35, 2|8, 3|7, 4|6.
%e A330759 T(10,3) = 10: 12|3|4, 13|2|4, 14|2|3, 1|23|4, 1|24|3, 1|2|34, 1|2|7, 1|3|6, 1|4|5, 2|3|5.
%e A330759 T(10,4) = 1: 1|2|3|4.
%e A330759 Triangle T(n,k) begins:
%e A330759   1;
%e A330759   0,  1;
%e A330759   0,  1;
%e A330759   0,  2,   1;
%e A330759   0,  2,   1;
%e A330759   0,  3,   2;
%e A330759   0,  4,   5,  1;
%e A330759   0,  5,   6,  1;
%e A330759   0,  6,   9,  2;
%e A330759   0,  8,  13,  3;
%e A330759   0, 10,  23, 10,  1;
%e A330759   0, 12,  27, 11,  1;
%e A330759   0, 15,  40, 19,  2;
%e A330759   0, 18,  51, 26,  3;
%e A330759   0, 22,  71, 40,  5;
%e A330759   0, 27, 100, 73, 16, 1;
%e A330759   ...
%p A330759 b:= proc(n, i, k) option remember; `if`(i*(i+1)/2<n, 0,
%p A330759       `if`(n=0, x^k, b(n, i-1, k) +(t-> b(n-i, t, k)*k
%p A330759         +b(n-i, t, k+1))(min(n-i, i-1))))
%p A330759     end:
%p A330759 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2, 0)):
%p A330759 seq(T(n), n=0..20);
%t A330759 b[n_, i_, k_] := b[n, i, k] = If[i(i+1)/2 < n, 0,
%t A330759      If[n == 0, x^k, b[n, i-1, k] + With[{t = Min[n-i, i-1]},
%t A330759      b[n-i, t, k]*k + b[n-i, t, k+1]]]];
%t A330759 T[n_] := CoefficientList[b[n, n, 0], x];
%t A330759 T /@ Range[0, 20] // Flatten (* _Jean-François Alcover_, Mar 12 2021, after _Alois P. Heinz_ *)
%Y A330759 Columns k=0-1 give: A000007, A000009 (for n>0).
%Y A330759 Row sums give A294617.
%Y A330759 Cf. A000041, A000096, A000217, A003056, A072706, A330460 (another version), A330765.
%K A330759 nonn,tabf
%O A330759 0,7
%A A330759 _Alois P. Heinz_, Dec 29 2019