%I #20 Apr 24 2016 05:36:19

%S 1,3,1,9,4,2,30,16,9,5,112,67,41,25,15,463,299,195,127,82,52,2095,

%T 1429,979,670,456,307,203,10279,7307,5204,3702,2623,1845,1283,877,

%U 54267,39848,29278,21485,15717,11437,8257,5894,4140,306298,230884,174029,131007,98367,73561,54692,40338,29427,21147

%N 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 Alois P. Heinz, <a href="/A270702/b270702.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) = A270701(n,n-k+1).

%e 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 Triangle T(n,k) begins:

%e : 1;

%e : 3, 1;

%e : 9, 4, 2;

%e : 30, 16, 9, 5;

%e : 112, 67, 41, 25, 15;

%e : 463, 299, 195, 127, 82, 52;

%e : 2095, 1429, 979, 670, 456, 307, 203;

%e : 10279, 7307, 5204, 3702, 2623, 1845, 1283, 877;

%e : 54267, 39848, 29278, 21485, 15717, 11437, 8257, 5894, 4140;

%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, 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, 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: A124427, A270765, A270766, A270767, A270768, A270769, A270770, A270771, A270772, A270773.

%Y Main and lower diagonals give: A000110(n-1), A270756, A270757, A270758, A270759, A270760, A270761, A270762, A270763, A270764.

%Y Row sums give A070071.

%Y Reflected triangle gives A270701.

%Y T(2n-1,n) gives A270703.

%K nonn,tabl

%O 1,2

%A _Alois P. Heinz_, Mar 21 2016