A258219 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.

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

Offset

0,5

Comments

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

Links

Alois P. Heinz, Antidiagonals n = 0..140, flattened

A. N. Stokes, Continued fraction solutions of the Riccati equation, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.

Wikipedia, Feynman diagram

Wikipedia, Lattice path

Formula

A(n,k) = Sum_{i=0..min(n,k)} C(k,i) * i! * A258220(n,i).

Example

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, ...
  ...

Maple

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);

Mathematica

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 *)

Crossrefs

Columns k=0-2 give: A005411 (for n>0), A000698(n+1), A005412(n+1).

Rows n=0-2 give: A000012, A000027(k+1), A028552(k+1).

Main diagonal gives A292693.

Cf. A258220, A258222.

Sequence in context: A121426 A190183 A004515 * A036560 A308244 A117297

Adjacent sequences: A258216 A258217 A258218 * A258220 A258221 A258222

Keyword

nonn,tabl

Author

Alois P. Heinz, May 23 2015

Status

approved