A355503 Total number of m-tuples (p_1, p_2, ..., p_m) of Dyck paths of semilength n-m, such that each p_i is never below p_{i-1} for m=0..n.

1, 2, 3, 5, 11, 35, 164, 1120, 10969, 152849, 3029650, 85227078, 3400752392, 192644205130, 15470939367651, 1761760468965521, 284641456742538865, 65175288287611738435, 21159611204475209730138, 9743708333490185603430830, 6357930817596444858142966826

Offset: 0

Author

Alois P. Heinz, Jul 04 2022

Keywords

• nonn

Examples

			a(3) = 5: ( ), (/\/\), (//\\), (/\, /\, /\), (<>, <>, <>, <>).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..115
• Wikipedia, Counting lattice paths

Crossrefs

Cf. A000108, A074962, A078920, A123352, A355400.

Antidiagonal sums of A368025.

Programs

Maple

a:= n-> add(mul(mul((i+j+2*(n-m))/(i+j), j=i..m-1), i=1..m-1), m=0..n):
seq(a(n), n=0..23);

Mathematica

Table[Sum[Product[Product[(i+j+2*(n-m))/(i+j), {j,i,m-1}], {i,1,m-1}], {m,0,n}], {n,0,20}] (* Vaclav Kotesovec, Aug 27 2023 *)
Table[Sum[BarnesG[1 + m] * Sqrt[BarnesG[1 + 2*n] * BarnesG[2 - 2*m + 2*n] * Gamma[1 + 2*m] * Gamma[1 + n] / (BarnesG[1 + 2*m] * Gamma[1 + m] * Gamma[1 + 2*n] * Gamma[1 - m + n])] / BarnesG[1 - m + 2*n], {m, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 27 2023 *)

Formula

a(n) = Sum_{m=0..n} Product_{i=1..m-1, j=i..m-1} (i+j+2*(n-m))/(i+j).

a(n) = 1 + Sum_{k=0..n-1} A078920(n-1,k).

a(n) = 1 + Sum_{k=0..n-1} A123352(n-1,k).

a(n) = Sum_{k=0..n} A368025(n-k, k).

From Vaclav Kotesovec, Aug 27 2023: (Start)

a(n) ~ c * exp(1/24) * 3^(n^2 - n/2) / (sqrt(A) * n^(1/24) * 2^((4*n^2-n-1)/3)), where A = A074962 is the Glaisher-Kinkelin constant and

c = Sum_{k,-oo,oo} 2^((k + mod(n,3)/3)/2 - 3*(k + mod(n,3)/3)^2/2).

Numerically, c = 1.78933741155287907159762028... if mod(n,3)=0 or mod(n,3)=1 and c = 1.78893263307672974352375161... if mod(n,3)=2. (End)