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 A238129 Triangle read by rows: T(n,k) gives the number of ballot sequences of length n having largest ascent k, n>=0, 0<=k<=n. 12
 1, 1, 0, 1, 1, 0, 1, 3, 0, 0, 1, 8, 1, 0, 0, 1, 19, 5, 1, 0, 0, 1, 47, 21, 6, 1, 0, 0, 1, 114, 78, 31, 7, 1, 0, 0, 1, 286, 292, 133, 43, 8, 1, 0, 0, 1, 723, 1028, 586, 215, 57, 9, 1, 0, 0, 1, 1869, 3691, 2453, 1073, 325, 73, 10, 1, 0, 0, 1, 4870, 13004, 10357, 5058, 1836, 467, 91, 11, 1, 0, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS Also number of standard Young tableaux with a pair of cells (v,v+1) such that v lies k rows below v+1, and no pair (u,u+1) with a larger such separation exists. LINKS Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 0..35, flattened EXAMPLE Triangle starts: 00: 1; 01: 1,     0; 02: 1,     1,     0; 03: 1,     3,     0,     0; 04: 1,     8,     1,     0,     0; 05: 1,    19,     5,     1,     0,    0; 06: 1,    47,    21,     6,     1,    0,    0; 07: 1,   114,    78,    31,     7,    1,    0,   0; 08: 1,   286,   292,   133,    43,    8,    1,   0,   0; 09: 1,   723,  1028,   586,   215,   57,    9,   1,   0,  0; 10: 1,  1869,  3691,  2453,  1073,  325,   73,  10,   1,  0, 0; 11: 1,  4870, 13004, 10357,  5058, 1836,  467,  91,  11,  1, 0, 0; 12: 1, 12943, 46452, 43462, 23953, 9631, 2941, 645, 111, 12, 1, 0, 0; ... MAPLE b:= proc(n, v, l) option remember; local m; m:=nops(l);       `if`(n<1, 1, expand(add(`if`(i=1 or l[i-1]>l[i],        (p->`if`(vadd(coeff(p, x, h)*`if`(h(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_List] := b[n, v, l] = Module[{m = Length[l]}, If[n<1, 1, Expand[Sum[If[i == 1 || l[[i-1]]>l[[i]], Function[{p}, If[v l[[i]]+1]]], 0], {i, 1, m}] + Function[{p}, Sum[Coefficient[p, x, h]*If[h

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Last modified October 18 03:19 EDT 2019. Contains 328135 sequences. (Running on oeis4.)