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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. 23
 1, 3, 1, 9, 4, 2, 30, 16, 9, 5, 112, 67, 41, 25, 15, 463, 299, 195, 127, 82, 52, 2095, 1429, 979, 670, 456, 307, 203, 10279, 7307, 5204, 3702, 2623, 1845, 1283, 877, 54267, 39848, 29278, 21485, 15717, 11437, 8257, 5894, 4140, 306298, 230884, 174029, 131007, 98367, 73561, 54692, 40338, 29427, 21147 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 LINKS Alois P. Heinz, Rows n = 1..141, flattened Wikipedia, Partition of a set FORMULA T(n,k) = A270701(n,n-k+1). EXAMPLE 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. Triangle T(n,k) begins: :      1; :      3,     1; :      9,     4,     2; :     30,    16,     9,     5; :    112,    67,    41,    25,    15; :    463,   299,   195,   127,    82,    52; :   2095,  1429,   979,   670,   456,   307,  203; :  10279,  7307,  5204,  3702,  2623,  1845, 1283,  877; :  54267, 39848, 29278, 21485, 15717, 11437, 8257, 5894, 4140; MAPLE b:= proc(n, m, t) option remember; `if`(n=0, [1, 0], add(      `if`(t=1 and j<>m+1, 0, (p->p+`if`(j=-t or t=1 and j=m+1,       [0, p[1]], 0))(b(n-1, max(m, j), `if`(t=1 and j=m+1, -j,      `if`(t<0, t, `if`(t>0, t-1, 0)))))), j=1..m+1))     end: T:= (n, k)-> b(n, 0, k)[2]: seq(seq(T(n, k), k=1..n), n=1..12); MATHEMATICA 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[n_, k_] := b[n, 0, k][[2]]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 12}] // Flatten (* Jean-François Alcover, Apr 24 2016, translated from Maple *) CROSSREFS Columns k=1-10 give: A124427, A270765, A270766, A270767, A270768, A270769, A270770, A270771, A270772, A270773. Main and lower diagonals give: A000110(n-1), A270756, A270757, A270758, A270759, A270760, A270761, A270762, A270763, A270764. Row sums give A070071. Reflected triangle gives A270701. T(2n-1,n) gives A270703. Sequence in context: A067417 A187887 A016577 * A124573 A127550 A021317 Adjacent sequences:  A270699 A270700 A270701 * A270703 A270704 A270705 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Mar 21 2016 STATUS approved

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Last modified January 18 04:40 EST 2019. Contains 319260 sequences. (Running on oeis4.)