|
%I #30 Aug 15 2022 10:28:35
%S 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,
%T 1638,8162,26368,1,7,54,425,3172,20898,110410,375733,1,8,70,630,5500,
%U 44164,307908,1708394,6092032,1,9,88,889,8838,82850,702844,5134293,29752066,110769550
%N 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.
%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.
%C 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
%H Alois P. Heinz, <a href="/A258219/b258219.txt">Antidiagonals n = 0..140, flattened</a>
%H A. N. Stokes, <a href="https://doi.org/10.1017/S0004972700005219">Continued fraction solutions of the Riccati equation</a>, Bull. Austral. Math. Soc. Vol. 25 (1982), 207-214.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Feynman_diagram">Feynman diagram</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! * A258220(n,i).
%e Square array A(n,k) begins:
%e 1, 1, 1, 1, 1, 1, ...
%e 1, 2, 3, 4, 5, 6, ...
%e 4, 10, 18, 28, 40, 54, ...
%e 25, 74, 153, 268, 425, 630, ...
%e 208, 706, 1638, 3172, 5500, 8838, ...
%e 2146, 8162, 20898, 44164, 82850, 143046, ...
%e ...
%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, (x+k*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, (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_ *)
%Y Columns k=0-2 give: A005411 (for n>0), A000698(n+1), A005412(n+1).
%Y Rows n=0-2 give: A000012, A000027(k+1), A028552(k+1).
%Y Main diagonal gives A292693.
%Y Cf. A258220, A258222.
%K nonn,tabl
%O 0,5
%A _Alois P. Heinz_, May 23 2015
|
