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%I #21 Jan 07 2015 05:41:00
%S 1,1,0,2,0,0,3,1,0,0,5,4,1,0,0,7,13,5,1,0,0,11,37,21,6,1,0,0,15,100,
%T 78,31,7,1,0,0,22,265,292,133,43,8,1,0,0,30,694,1028,586,215,57,9,1,0,
%U 0,42,1828,3691,2453,1073,325,73,10,1,0,0,56,4815,13004,10357,5058,1836,467,91,11,1,0,0
%N Triangle read by rows: T(n,k) gives the number of ballot sequences of length n having largest descent k, n>=0, 0<=k<=n.
%C Also number of standard Young tableaux with a pair of cells (v,v+1) such that v lies k rows above v+1, and no pair (u,u+1) with a larger such separation exists.
%H Joerg Arndt and Alois P. Heinz, <a href="/A238128/b238128.txt">Table of n, a(n) for n = 0..35, flattened</a>
%e Triangle starts:
%e 00: 1;
%e 01: 1, 0;
%e 02: 2, 0, 0;
%e 03: 3, 1, 0, 0;
%e 04: 5, 4, 1, 0, 0;
%e 05: 7, 13, 5, 1, 0, 0;
%e 06: 11, 37, 21, 6, 1, 0, 0;
%e 07: 15, 100, 78, 31, 7, 1, 0, 0;
%e 08: 22, 265, 292, 133, 43, 8, 1, 0, 0;
%e 09: 30, 694, 1028, 586, 215, 57, 9, 1, 0, 0;
%e 10: 42, 1828, 3691, 2453, 1073, 325, 73, 10, 1, 0, 0;
%e 11: 56, 4815, 13004, 10357, 5058, 1836, 467, 91, 11, 1, 0, 0;
%e 12: 77, 12867, 46452, 43462, 23953, 9631, 2941, 645, 111, 12, 1, 0, 0;
%e ...
%p b:= proc(n, v, l) option remember; `if`(n<1, 1, expand(add(
%p `if`(i=1 or l[i-1]>l[i], (p->`if`(i<v, add(coeff(p, x, h)*
%p `if`(h<v-i, x^(v-i), x^h), h=0..max(v-i, degree(p))), p))
%p (b(n-1, i, subsop(i=l[i]+1, l))), 0), i=1..nops(l))+
%p b(n-1, nops(l)+1, [l[], 1])))
%p end:
%p T:= n-> (p->seq(coeff(p, x, i), i=0..n))(b(n-1, 1, [1])):
%p seq(T(n), n=0..12);
%t b[n_, v_, l_List] := b[n, v, l] = If[n<1, 1, Expand[Sum[If[i == 1 || l[[i-1]] > l[[i]], Function[{p}, If[i<v, Sum[Coefficient[p, x, h]* If[h < v-i, x^(v-i), x^h], {h, 0, Max[v-i, Exponent[p, x]]}], p]][b[n-1, i, ReplacePart[l, 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, 12}] // Flatten (* _Jean-François Alcover_, Jan 07 2015, translated from Maple *)
%Y Columns k=0-10 give: A000041, A244197, A244198, A244199, A244200, A244201, A244202, A244203, A244204, A244205, A244206.
%Y Row sums are A000085.
%Y Cf. A238129.
%K nonn,tabl
%O 0,4
%A _Joerg Arndt_ and _Alois P. Heinz_, Feb 21 2014
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