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A270702 Total sum T(n,k) of the sizes of all blocks with minimal element k in all set partitions of {1,2,...,n}; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

%I A270702 #20 Apr 24 2016 05:36:19
%S A270702 1,3,1,9,4,2,30,16,9,5,112,67,41,25,15,463,299,195,127,82,52,2095,
%T A270702 1429,979,670,456,307,203,10279,7307,5204,3702,2623,1845,1283,877,
%U A270702 54267,39848,29278,21485,15717,11437,8257,5894,4140,306298,230884,174029,131007,98367,73561,54692,40338,29427,21147
%N A270702 Total sum T(n,k) of the sizes of all blocks with minimal element k in all set partitions of {1,2,...,n}; triangle T(n,k), n>=1, 1<=k<=n, read by rows.
%H A270702 Alois P. Heinz, <a href="/A270702/b270702.txt">Rows n = 1..141, flattened</a>
%H A270702 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A270702 T(n,k) = A270701(n,n-k+1).
%e A270702 Row n=3 is [9, 4, 2] = [3+2+2+1+1, 0+0+1+2+1, 0+1+0+0+1] because the set partitions of {1,2,3} are: 123, 12|3, 13|2, 1|23, 1|2|3.
%e A270702 Triangle T(n,k) begins:
%e A270702 :      1;
%e A270702 :      3,     1;
%e A270702 :      9,     4,     2;
%e A270702 :     30,    16,     9,     5;
%e A270702 :    112,    67,    41,    25,    15;
%e A270702 :    463,   299,   195,   127,    82,    52;
%e A270702 :   2095,  1429,   979,   670,   456,   307,  203;
%e A270702 :  10279,  7307,  5204,  3702,  2623,  1845, 1283,  877;
%e A270702 :  54267, 39848, 29278, 21485, 15717, 11437, 8257, 5894, 4140;
%p A270702 b:= proc(n, m, t) option remember; `if`(n=0, [1, 0], add(
%p A270702      `if`(t=1 and j<>m+1, 0, (p->p+`if`(j=-t or t=1 and j=m+1,
%p A270702       [0, p[1]], 0))(b(n-1, max(m, j), `if`(t=1 and j=m+1, -j,
%p A270702      `if`(t<0, t, `if`(t>0, t-1, 0)))))), j=1..m+1))
%p A270702     end:
%p A270702 T:= (n, k)-> b(n, 0, k)[2]:
%p A270702 seq(seq(T(n, k), k=1..n), n=1..12);
%t A270702 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 A270702 T[n_, k_] := b[n, 0, k][[2]];
%t A270702 Table[Table[T[n, k], {k, 1, n}], {n, 1, 12}] // Flatten (* _Jean-François Alcover_, Apr 24 2016, translated from Maple *)
%Y A270702 Columns k=1-10 give: A124427, A270765, A270766, A270767, A270768, A270769, A270770, A270771, A270772, A270773.
%Y A270702 Main and lower diagonals give: A000110(n-1), A270756, A270757, A270758, A270759, A270760, A270761, A270762, A270763, A270764.
%Y A270702 Row sums give A070071.
%Y A270702 Reflected triangle gives A270701.
%Y A270702 T(2n-1,n) gives A270703.
%K A270702 nonn,tabl
%O A270702 1,2
%A A270702 _Alois P. Heinz_, Mar 21 2016