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%I #31 May 23 2018 02:42:47
%S 1,4,2,17,10,3,76,52,18,4,362,274,111,28,5,1842,1500,675,200,40,6,
%T 9991,8614,4185,1380,325,54,7,57568,51992,26832,9568,2510,492,70,8,
%U 351125,329650,178755,67820,19255,4206,707,88,9,2259302,2192434,1239351,494828,149605,35382,6629,976,108,10
%N Sum T(n,k) of the k-th entries in all blocks of all set partitions of [n]; triangle T(n,k), n>=1, 1<=k<=n, read by rows.
%C T(n,k) is also k times the number of blocks of size >k in all set partitions of [n+1]. T(3,2) = 10 = 2 * 5 because there are 5 blocks of size >2 in all set partitions of [4], namely in 1234, 123|4, 124|3, 134|2, 1|234.
%H Alois P. Heinz, <a href="/A285595/b285595.txt">Rows n = 1..141, flattened</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F T(n,k) = k * Sum_{j=k+1..n+1} binomial(n+1,j)*A000110(n+1-j).
%F T(n,k) = k * Sum_{j=k+1..n+1} A175757(n+1,j).
%F Sum_{k=1..n} T(n,k)/k = A278677(n+1).
%e T(3,2) = 10 because the sum of the second entries in all blocks of all set partitions of [3] (123, 12|3, 13|2, 1|23, 1|2|3) is 2+2+3+3+0 = 10.
%e Triangle T(n,k) begins:
%e : 1;
%e : 4, 2;
%e : 17, 10, 3;
%e : 76, 52, 18, 4;
%e : 362, 274, 111, 28, 5;
%e : 1842, 1500, 675, 200, 40, 6;
%e : 9991, 8614, 4185, 1380, 325, 54, 7;
%e : 57568, 51992, 26832, 9568, 2510, 492, 70, 8;
%p T:= proc(h) option remember; local b; b:=
%p proc(n, l) option remember; `if`(n=0, [1, 0],
%p (p-> p+[0, (h-n+1)*p[1]*x^1])(b(n-1, [l[], 1]))+
%p add((p-> p+[0, (h-n+1)*p[1]*x^(l[j]+1)])(b(n-1,
%p sort(subsop(j=l[j]+1, l), `>`))), j=1..nops(l)))
%p end: (p-> seq(coeff(p, x, i), i=1..n))(b(h, [])[2])
%p end:
%p seq(T(n), n=1..12);
%p # second Maple program:
%p b:= proc(n) option remember; `if`(n=0, [1, 0],
%p add((p-> p+[0, p[1]*add(x^k, k=1..j-1)])(
%p b(n-j)*binomial(n-1, j-1)), j=1..n))
%p end:
%p T:= n-> (p-> seq(coeff(p, x, i)*i, i=1..n))(b(n+1)[2]):
%p seq(T(n), n=1..12);
%t b[n_] := b[n] = If[n == 0, {1, 0}, Sum[# + {0, #[[1]]*Sum[x^k, {k, 1, j-1} ]}&[b[n - j]*Binomial[n - 1, j - 1]], {j, 1, n}]];
%t T[n_] := Table[Coefficient[#, x, i]*i, {i, 1, n}] &[b[n + 1][[2]]];
%t Table[T[n], {n, 1, 12}] // Flatten (* _Jean-François Alcover_, May 23 2018, translated from 2nd Maple program *)
%Y Column k=1 gives A124325(n+1).
%Y Row sums give A000110(n) * A000217(n) = A105488(n+3).
%Y Main diagonal and first lower diagonal give: A000027, A028552.
%Y Cf. A007318, A175757, A278677, A283424, A285362, A285793, A286897.
%K nonn,tabl
%O 1,2
%A _Alois P. Heinz_, Apr 22 2017
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