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%I #21 Sep 20 2017 19:14:24
%S 1,1,1,1,2,2,1,3,10,5,1,4,24,74,14,1,5,44,297,706,42,1,6,70,764,4896,
%T 8162,132,1,7,102,1565,17924,100278,110410,429,1,8,140,2790,47650,
%U 527844,2450304,1708394,1430,1,9,184,4529,104454,1831250,18685164,69533397,29752066,4862
%N A(n,k) is the sum over all Dyck paths of semilength n of products over all peaks p of (k*x_p+y_p)/y_p, where x_p and y_p are the coordinates of peak p; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%C A Dyck path of semilength n is a (x,y)-lattice path from (0,0) to (2n,0) that does not go below the x-axis and consists of steps U=(1,1) and D=(1,-1). A peak of a Dyck path is any lattice point visited between two consecutive steps UD.
%H Alois P. Heinz, <a href="/A258222/b258222.txt">Antidiagonals n = 0..140, flattened</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path">Lattice path</a>
%F A(n,k) = Sum_{i=0..min(n,k)} C(k,i) * i! * A258223(n,i).
%e Square array A(n,k) begins:
%e : 1, 1, 1, 1, 1, 1, ...
%e : 1, 2, 3, 4, 5, 6, ...
%e : 2, 10, 24, 44, 70, 102, ...
%e : 5, 74, 297, 764, 1565, 2790, ...
%e : 14, 706, 4896, 17924, 47650, 104454, ...
%e : 42, 8162, 100278, 527844, 1831250, 4953222, ...
%p b:= proc(x, y, t, k) option remember; `if`(y>x or y<0, 0,
%p `if`(x=0, 1, b(x-1, y-1, false, k)*`if`(t, (k*x+y)/y, 1)
%p + b(x-1, y+1, true, k) ))
%p end:
%p A:= (n, k)-> b(2*n, 0, false, k):
%p seq(seq(A(n, d-n), n=0..d), d=0..12);
%t b[x_, y_, t_, k_] := b[x, y, t, k] = If[y > x || y < 0, 0, If[x == 0, 1, b[x - 1, y - 1, False, k]*If[t, (k*x + y)/y, 1] + b[x - 1, y + 1, True, k]]];
%t A [n_, k_] := b[2*n, 0, False, k];
%t Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* _Jean-François Alcover_, Apr 23 2016, translated from Maple *)
%Y Columns k=0-1 give: A000108, A000698(n+1).
%Y Rows n=0-2 give: A000012, A000027(k+1), A049450(k+1).
%Y Main diagonal gives A292694.
%Y Cf. A258219, A258223.
%K nonn,tabl
%O 0,5
%A _Alois P. Heinz_, May 23 2015
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