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A350647 Number T(n,k) of partitions of [n] having k blocks containing their own index when blocks are ordered with decreasing largest elements; triangle T(n,k), n>=0, 0<=k<=ceiling(n/2), read by rows.

%I A350647 #25 Jan 11 2022 18:31:44
%S A350647 1,0,1,1,1,1,3,1,6,7,2,16,25,10,1,73,91,35,4,298,390,163,25,1,1453,
%T A350647 1797,755,128,7,7366,9069,3919,737,55,1,40689,49106,21485,4304,380,11,
%U A350647 238258,284537,126273,26695,2696,110,1,1483306,1751554,785435,173038,19272,976,16
%N A350647 Number T(n,k) of partitions of [n] having k blocks containing their own index when blocks are ordered with decreasing largest elements; triangle T(n,k), n>=0, 0<=k<=ceiling(n/2), read by rows.
%H A350647 Alois P. Heinz, <a href="/A350647/b350647.txt">Rows n = 0..200, flattened</a>
%H A350647 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A350647 Sum_{k=1..ceiling(n/2)} k * T(n,k) = A350648(n).
%e A350647 T(4,0) = 6: 432|1, 42|31, 42|3|1, 4|31|2, 4|3|21, 4|3|2|1.
%e A350647 T(4,1) = 7: 4321, 43|21, 43|2|1, 421|3, 4|321, 4|32|1, 41|3|2.
%e A350647 T(4,2) = 2: 431|2, 41|32.
%e A350647 T(5,2) = 10: 5431|2, 541|32, 531|42, 51|432, 521|4|3, 5|421|3, 5|42|31, 5|42|3|1, 51|4|32, 51|4|3|2.
%e A350647 T(5,3) = 1: 51|42|3.
%e A350647 Triangle T(n,k) begins:
%e A350647        1;
%e A350647        0,      1;
%e A350647        1,      1;
%e A350647        1,      3,      1;
%e A350647        6,      7,      2;
%e A350647       16,     25,     10,     1;
%e A350647       73,     91,     35,     4;
%e A350647      298,    390,    163,    25,    1;
%e A350647     1453,   1797,    755,   128,    7;
%e A350647     7366,   9069,   3919,   737,   55,   1;
%e A350647    40689,  49106,  21485,  4304,  380,  11;
%e A350647   238258, 284537, 126273, 26695, 2696, 110, 1;
%e A350647   ...
%p A350647 b:= proc(n, m) option remember; expand(`if`(n=0, 1, add(
%p A350647       `if`(j=n, x, 1)*b(n-1, max(m, j)), j=1..m+1)))
%p A350647     end:
%p A350647 T:= n-> (p-> seq(coeff(p, x, i), i=0..ceil(n/2)))(b(n, 0)):
%p A350647 seq(T(n), n=0..14);
%t A350647 b[n_, m_] := b[n, m] = Expand[If[n == 0, 1, Sum[
%t A350647      If[j == n, x, 1]*b[n-1, Max[m, j]], {j, 1, m+1}]]];
%t A350647 T[n_] := With[{p = b[n, 0]},
%t A350647 Table[Coefficient[p, x, i], {i, 0, Ceiling[n/2]}]];
%t A350647 Table[T[n], {n, 0, 14}] // Flatten (* _Jean-François Alcover_, Jan 11 2022, after _Alois P. Heinz_ *)
%Y A350647 Columns k=0-1 give: A350649, A350650.
%Y A350647 Row sums give A000110.
%Y A350647 T(2n,n) gives A000124(n-1) for n>=1.
%Y A350647 T(2n+1,n+1) gives A000012.
%Y A350647 Cf. A259691, A350648.
%K A350647 nonn,tabf
%O A350647 0,7
%A A350647 _Alois P. Heinz_, Jan 09 2022