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 A238121 Triangle read by rows: T(n,k) gives the number of ballot sequences of length n having exactly k descents, n>=0, 0<=k<=n. 14
 1, 1, 0, 2, 0, 0, 3, 1, 0, 0, 5, 5, 0, 0, 0, 7, 16, 3, 0, 0, 0, 11, 43, 21, 1, 0, 0, 0, 15, 99, 101, 17, 0, 0, 0, 0, 22, 215, 373, 145, 9, 0, 0, 0, 0, 30, 430, 1174, 836, 146, 4, 0, 0, 0, 0, 42, 834, 3337, 3846, 1324, 112, 1, 0, 0, 0, 0, 56, 1529, 8642, 15002, 8786, 1615, 66, 0, 0, 0, 0, 0, 77, 2765, 21148, 52132, 47013, 15403, 1582, 32, 0, 0, 0, 0, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Also number of standard Young tableaux such that there are k pairs of cells (v,v+1) with v+1 lying in a row above v. Columns k=0-10 give: A000041, A241794, A241795, A241796, A241797, A241798, A241799, A241800, A241801, A241802, A241803. T(2n,n) gives A241804. T(2n+1,n) gives A241805. Row sums are A000085. T(n*(n+1)/2,n*(n-1)/2) = 1. A238122 is another version with zeros omitted. LINKS Joerg Arndt and Alois P. Heinz, Rows n = 0..45, flattened EXAMPLE Triangle starts: 1, 1, 0, 2, 0, 0, 3, 1, 0, 0, 5, 5, 0, 0, 0, 7, 16, 3, 0, 0, 0, 11, 43, 21, 1, 0, 0, 0, 15, 99, 101, 17, 0, 0, 0, 0, 22, 215, 373, 145, 9, 0, 0, 0, 0, 30, 430, 1174, 836, 146, 4, 0, 0, 0, 0, 42, 834, 3337, 3846, 1324, 112, 1, 0, 0, 0, 0, 56, 1529, 8642, 15002, 8786, 1615, 66, 0, 0, 0, 0, 0, 77, 2765, 21148, 52132, 47013, 15403, 1582, 32, 0, 0, 0, 0, 0, 101, 4792, 48713, 164576, 214997, 112106, 21895, 1310, 14, 0, 0, 0, 0, 0, ... The T(5,1) = 16 ballot sequences of length n=5 with k=1 descent are (dots for zeros): 01:  [ . . . 1 . ] 02:  [ . . 1 . . ] 03:  [ . . 1 . 1 ] 04:  [ . . 1 . 2 ] 05:  [ . . 1 1 . ] 06:  [ . . 1 2 . ] 07:  [ . . 1 2 1 ] 08:  [ . 1 . . . ] 09:  [ . 1 . . 1 ] 10:  [ . 1 . . 2 ] 11:  [ . 1 . 1 2 ] 12:  [ . 1 . 2 3 ] 13:  [ . 1 2 . . ] 14:  [ . 1 2 . 1 ] 15:  [ . 1 2 . 3 ] 16:  [ . 1 2 3 . ] MAPLE b:= proc(n, v, l) option remember; `if`(n<1, 1, expand(       add(`if`(i=1 or l[i-1]>l[i], `if`(i (p-> seq(coeff(p, x, i), i=0..n))(b(n-1, 1, [1])): seq(T(n), n=0..14); MATHEMATICA b[n_, v_, l_] := b[n, v, l] = If[n<1, 1, Sum[If[i == 1 || l[[i-1]] > l[[i]], If[i l[[i]]+1]], 0], {i, 1, Length[l]}] + b[n-1, Length[l]+1, Append[l, 1]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, n}]][b[n-1, 1, {1}]]; Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Jan 06 2015, translated from Maple *) CROSSREFS Sequence in context: A259479 A238343 A238128 * A171380 A323592 A170980 Adjacent sequences:  A238118 A238119 A238120 * A238122 A238123 A238124 KEYWORD nonn,tabl AUTHOR Joerg Arndt and Alois P. Heinz, Feb 21 2014 STATUS approved

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Last modified July 24 21:47 EDT 2021. Contains 346273 sequences. (Running on oeis4.)