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A258219
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A(n,k) is the sum over all Dyck paths of semilength n of products over all peaks p of (x_p+k*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.
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7
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1, 1, 1, 1, 2, 4, 1, 3, 10, 25, 1, 4, 18, 74, 208, 1, 5, 28, 153, 706, 2146, 1, 6, 40, 268, 1638, 8162, 26368, 1, 7, 54, 425, 3172, 20898, 110410, 375733, 1, 8, 70, 630, 5500, 44164, 307908, 1708394, 6092032, 1, 9, 88, 889, 8838, 82850, 702844, 5134293, 29752066, 110769550
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OFFSET
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0,5
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COMMENTS
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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.
Conjecture: the g.f. G(k,x) for the k-th column satisfies the Riccati differential equation 2*x^2*d/dx(G(k,x)) + 1 + (k*x - 1)*G(k,x) + x*G^2(k,x) = 0 and hence, by Stokes 1982, has the continued fraction representation G(k,x) = 1/(1 - (k+1)*x/(1 - 3*x/(1 - (k+3)*x/(1 - 5*x/(1 - (k+5)*x/(1 - 7*x/(1 - ...))))))) of Stieltjes type. - Peter Bala, Jul 28 2022
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LINKS
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FORMULA
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A(n,k) = Sum_{i=0..min(n,k)} C(k,i) * i! * A258220(n,i).
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EXAMPLE
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Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, ...
1, 2, 3, 4, 5, 6, ...
4, 10, 18, 28, 40, 54, ...
25, 74, 153, 268, 425, 630, ...
208, 706, 1638, 3172, 5500, 8838, ...
2146, 8162, 20898, 44164, 82850, 143046, ...
...
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MAPLE
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b:= proc(x, y, t, k) option remember; `if`(y>x or y<0, 0,
`if`(x=0, 1, b(x-1, y-1, false, k)*`if`(t, (x+k*y)/y, 1)
+ b(x-1, y+1, true, k) ))
end:
A:= (n, k)-> b(2*n, 0, false, k):
seq(seq(A(n, d-n), n=0..d), d=0..12);
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MATHEMATICA
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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, (x+k*y)/y, 1] + b[x-1, y+1, True, k]]]; A[n_, k_] := b[2*n, 0, False, k]; Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Jan 09 2016, after Alois P. Heinz *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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