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 A270701 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. 23
 1, 1, 3, 2, 4, 9, 5, 9, 16, 30, 15, 25, 41, 67, 112, 52, 82, 127, 195, 299, 463, 203, 307, 456, 670, 979, 1429, 2095, 877, 1283, 1845, 2623, 3702, 5204, 7307, 10279, 4140, 5894, 8257, 11437, 15717, 21485, 29278, 39848, 54267, 21147, 29427, 40338, 54692, 73561, 98367, 131007, 174029, 230884, 306298 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 LINKS Alois P. Heinz, Rows n = 1..141, flattened Wikipedia, Partition of a set FORMULA T(n,k) = A270702(n,n-k+1). EXAMPLE 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. Triangle T(n,k) begins: :     1; :     1,    3; :     2,    4,    9; :     5,    9,   16,    30; :    15,   25,   41,    67,   112; :    52,   82,  127,   195,   299,   463; :   203,  307,  456,   670,   979,  1429,  2095; :   877, 1283, 1845,  2623,  3702,  5204,  7307, 10279; :  4140, 5894, 8257, 11437, 15717, 21485, 29278, 39848, 54267; 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, max(0, 1+n-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, Max[0, 1+n-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: A000110(n-1), A270756, A270757, A270758, A270759, A270760, A270761, A270762, A270763, A270764. Main and lower diagonals give: A124427, A270765, A270766, A270767, A270768, A270769, A270770, A270771, A270772, A270773. Row sums give A070071. Reflected triangle gives A270702. T(2n-1,n) gives A270703. Sequence in context: A201838 A099257 A325909 * A083762 A173028 A264985 Adjacent sequences:  A270698 A270699 A270700 * A270702 A270703 A270704 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Mar 21 2016 STATUS approved

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Last modified October 15 10:15 EDT 2019. Contains 328026 sequences. (Running on oeis4.)