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A285362 Sum T(n,k) of the entries in the k-th blocks of all set partitions of [n]; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

%I A285362 #25 Aug 21 2021 06:36:41
%S A285362 1,4,2,15,12,3,60,58,28,4,262,273,185,55,5,1243,1329,1094,495,96,6,
%T A285362 6358,6839,6293,3757,1148,154,7,34835,37423,36619,26421,11122,2380,
%U A285362 232,8,203307,217606,219931,180482,96454,28975,4518,333,9,1257913,1340597,1376929,1230737,787959,308127,67898,7995,460,10
%N A285362 Sum T(n,k) of the entries in the k-th blocks of all set partitions of [n]; triangle T(n,k), n>=1, 1<=k<=n, read by rows.
%H A285362 Alois P. Heinz, <a href="/A285362/b285362.txt">Rows n = 1..141, flattened</a>
%H A285362 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%e A285362 T(3,2) = 12 because the sum of the entries in the second blocks of all set partitions of [3] (123, 12|3, 13|2, 1|23, 1|2|3) is 0+3+2+5+2 = 12.
%e A285362 Triangle T(n,k) begins:
%e A285362       1;
%e A285362       4,     2;
%e A285362      15,    12,     3;
%e A285362      60,    58,    28,     4;
%e A285362     262,   273,   185,    55,     5;
%e A285362    1243,  1329,  1094,   495,    96,    6;
%e A285362    6358,  6839,  6293,  3757,  1148,  154,   7;
%e A285362   34835, 37423, 36619, 26421, 11122, 2380, 232, 8;
%e A285362   ...
%p A285362 T:= proc(h) option remember; local b; b:=
%p A285362       proc(n, m) option remember; `if`(n=0, [1, 0], add((p-> p
%p A285362         +[0, (h-n+1)*p[1]*x^j])(b(n-1, max(m, j))), j=1..m+1))
%p A285362       end: (p-> seq(coeff(p, x, i), i=1..n))(b(h, 0)[2])
%p A285362     end:
%p A285362 seq(T(n), n=1..12);
%t A285362 T[h_] := T[h] = Module[{b}, b[n_, m_] := b[n, m] = If[n == 0, {1, 0}, Sum[# + {0, (h - n + 1)*#[[1]]*x^j}&[b[n - 1, Max[m, j]]], {j, 1, m + 1}]]; Table[Coefficient[#, x, i], {i, 1, n}]&[b[h, 0][[2]]]];
%t A285362 Table[T[n], {n, 1, 12}] // Flatten (* _Jean-François Alcover_, Apr 30 2018, after _Alois P. Heinz_ *)
%Y A285362 Columns k=1-10 give: A285363, A285364, A285365, A285366, A285367, A285368, A285369, A285370, A285371, A285372.
%Y A285362 Row sums give A000110(n) * A000217(n) = A105488(n+3).
%Y A285362 Main diagonal and first lower diagonal give: A000027, A006000 (for n>0).
%Y A285362 T(2n+1,n+1) gives A285410.
%Y A285362 Cf. A270236, A285439, A285595.
%K A285362 nonn,tabl
%O A285362 1,2
%A A285362 _Alois P. Heinz_, Apr 17 2017