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%I #22 Apr 24 2016 05:36:34
%S 1,1,3,2,4,9,5,9,16,30,15,25,41,67,112,52,82,127,195,299,463,203,307,
%T 456,670,979,1429,2095,877,1283,1845,2623,3702,5204,7307,10279,4140,
%U 5894,8257,11437,15717,21485,29278,39848,54267,21147,29427,40338,54692,73561,98367,131007,174029,230884,306298
%N Total sum T(n,k) of the sizes of all blocks with maximal element k in all set partitions of {1,2,...,n}; triangle T(n,k), n>=1, 1<=k<=n, read by rows.
%H Alois P. Heinz, <a href="/A270701/b270701.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) = A270702(n,n-k+1).
%e Row n=3 is [2, 4, 9] = [0+0+0+1+1, 0+2+1+0+1, 3+1+2+2+1] because the set partitions of {1,2,3} are: 123, 12|3, 13|2, 1|23, 1|2|3.
%e Triangle T(n,k) begins:
%e : 1;
%e : 1, 3;
%e : 2, 4, 9;
%e : 5, 9, 16, 30;
%e : 15, 25, 41, 67, 112;
%e : 52, 82, 127, 195, 299, 463;
%e : 203, 307, 456, 670, 979, 1429, 2095;
%e : 877, 1283, 1845, 2623, 3702, 5204, 7307, 10279;
%e : 4140, 5894, 8257, 11437, 15717, 21485, 29278, 39848, 54267;
%p b:= proc(n, m, t) option remember; `if`(n=0, [1, 0], add(
%p `if`(t=1 and j<>m+1, 0, (p->p+`if`(j=-t or t=1 and j=m+1,
%p [0, p[1]], 0))(b(n-1, max(m, j), `if`(t=1 and j=m+1, -j,
%p `if`(t<0, t, `if`(t>0, t-1, 0)))))), j=1..m+1))
%p end:
%p T:= (n, k)-> b(n, 0, max(0, 1+n-k))[2]:
%p seq(seq(T(n, k), k=1..n), n=1..12);
%t b[n_, m_, t_] := b[n, m, t] = If[n == 0, {1, 0}, Sum[If[t == 1 && j != m+1, 0, Function[p, p + If[j == -t || t == 1 && j == m+1, {0, p[[1]]}, 0]][b[ n-1, Max[m, j], If[t == 1 && j == m+1, -j, If[t < 0, t, If[t > 0, t-1, 0] ]]]]], {j, 1, m+1}]];
%t T[n_, k_] := b[n, 0, Max[0, 1+n-k]][[2]];
%t Table[Table[T[n, k], {k, 1, n}], {n, 1, 12}] // Flatten (* _Jean-François Alcover_, Apr 24 2016, translated from Maple *)
%Y Columns k=1-10 give: A000110(n-1), A270756, A270757, A270758, A270759, A270760, A270761, A270762, A270763, A270764.
%Y Main and lower diagonals give: A124427, A270765, A270766, A270767, A270768, A270769, A270770, A270771, A270772, A270773.
%Y Row sums give A070071.
%Y Reflected triangle gives A270702.
%Y T(2n-1,n) gives A270703.
%K nonn,tabl
%O 1,3
%A _Alois P. Heinz_, Mar 21 2016
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