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A238122 Irregular triangle read by rows: T(n,k) gives the number of ballot sequences of length n having k descents, n>=0, 0<=k<=A083920(n-1).

%I A238122 #25 Sep 26 2023 11:35:50
%S A238122 1,1,2,3,1,5,5,7,16,3,11,43,21,1,15,99,101,17,22,215,373,145,9,30,430,
%T A238122 1174,836,146,4,42,834,3337,3846,1324,112,1,56,1529,8642,15002,8786,
%U A238122 1615,66,77,2765,21148,52132,47013,15403,1582,32,101,4792,48713,164576,214997,112106,21895,1310,14
%N A238122 Irregular triangle read by rows: T(n,k) gives the number of ballot sequences of length n having k descents, n>=0, 0<=k<=A083920(n-1).
%C A238122 Same as A238121, with zeros omitted.
%C A238122 Columns k=0-10 give: A000041, A241794, A241795, A241796, A241797, A241798, A241799, A241800, A241801, A241802, A241803.
%C A238122 T(2n,n) gives A241804.
%C A238122 T(2n+1,n) gives A241805.
%C A238122 Row sums are A000085.
%C A238122 T(n*(n+1)/2,n*(n-1)/2) = 1.
%H A238122 Joerg Arndt and Alois P. Heinz, <a href="/A238122/b238122.txt">Rows n = 0..50, flattened</a>
%e A238122 T(5,0) = 7: [1,1,1,1,1], [1,1,1,1,2], [1,1,1,2,2], [1,1,1,2,3], [1,1,2,2,3], [1,1,2,3,4], [1,2,3,4,5].
%e A238122 T(5,1) = 16: [1,1,1,2,1], [1,1,2,1,1], [1,1,2,1,2], [1,1,2,1,3], [1,1,2,2,1], [1,1,2,3,1], [1,1,2,3,2], [1,2,1,1,1], [1,2,1,1,2], [1,2,1,1,3], [1,2,1,2,3], [1,2,1,3,4], [1,2,3,1,1], [1,2,3,1,2], [1,2,3,1,4], [1,2,3,4,1].
%e A238122 T(5,2) = 3: [1,2,1,2,1], [1,2,1,3,1], [1,2,1,3,2].
%e A238122 Triangle starts:
%e A238122 00:   1;
%e A238122 01:   1;
%e A238122 02:   2;
%e A238122 03:   3,    1;
%e A238122 04:   5,    5;
%e A238122 05:   7,   16,      3;
%e A238122 06:  11,   43,     21,      1;
%e A238122 07:  15,   99,    101,     17;
%e A238122 08:  22,  215,    373,    145,      9;
%e A238122 09:  30,  430,   1174,    836,    146,      4;
%e A238122 10:  42,  834,   3337,   3846,   1324,    112,      1;
%e A238122 11:  56, 1529,   8642,  15002,   8786,   1615,     66;
%e A238122 12:  77, 2765,  21148,  52132,  47013,  15403,   1582,    32;
%e A238122 13: 101, 4792,  48713, 164576, 214997, 112106,  21895,  1310,  14;
%e A238122 14: 135, 8216, 108147, 484609, 874413, 672015, 215849, 26159, 932, 5;
%e A238122 ...
%p A238122 b:= proc(n, v, l) option remember; `if`(n<1, 1, expand(
%p A238122       add(`if`(i=1 or l[i-1]>l[i], `if`(i<v, x, 1)*
%p A238122       b(n-1, i, subsop(i=l[i]+1, l)), 0), i=1..nops(l))+
%p A238122       b(n-1, nops(l)+1, [l[], 1])))
%p A238122     end:
%p A238122 T:= n->(p->seq(coeff(p, x, i), i=0..degree(p)))(b(n-1, 1, [1])):
%p A238122 seq(T(n), n=0..14);
%t A238122 b[n_, v_, l_List] := b[n, v, l] = If[n<1, 1, Expand[Sum[If[i == 1 || l[[i-1]] > l[[i]], If[i<v, x, 1]*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, Exponent[p, x]}]][b[n-1, 1, {1}]]; Table[T[n], {n, 0, 14}] // Flatten (* _Jean-François Alcover_, Feb 11 2015, after Maple *)
%K A238122 nonn,tabf
%O A238122 0,3
%A A238122 _Joerg Arndt_ and _Alois P. Heinz_, Feb 21 2014