cp's OEIS Frontend

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.

%I A355503 #27 Nov 16 2024 17:27:32
%S A355503 1,2,3,5,11,35,164,1120,10969,152849,3029650,85227078,3400752392,
%T A355503 192644205130,15470939367651,1761760468965521,284641456742538865,
%U A355503 65175288287611738435,21159611204475209730138,9743708333490185603430830,6357930817596444858142966826
%N 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.
%H A355503 Alois P. Heinz, <a href="/A355503/b355503.txt">Table of n, a(n) for n = 0..115</a>
%H A355503 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%F A355503 a(n) = Sum_{m=0..n} Product_{i=1..m-1, j=i..m-1} (i+j+2*(n-m))/(i+j).
%F A355503 a(n) = 1 + Sum_{k=0..n-1} A078920(n-1,k).
%F A355503 a(n) = 1 + Sum_{k=0..n-1} A123352(n-1,k).
%F A355503 a(n) = Sum_{k=0..n} A368025(n-k, k).
%F A355503 From _Vaclav Kotesovec_, Aug 27 2023: (Start)
%F A355503 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
%F A355503 c = Sum_{k,-oo,oo} 2^((k + mod(n,3)/3)/2 - 3*(k + mod(n,3)/3)^2/2).
%F A355503 Numerically, c = 1.78933741155287907159762028... if mod(n,3)=0 or mod(n,3)=1 and c = 1.78893263307672974352375161... if mod(n,3)=2. (End)
%e A355503 a(3) = 5: ( ), (/\/\), (//\\), (/\, /\, /\), (<>, <>, <>, <>).
%p A355503 a:= n-> add(mul(mul((i+j+2*(n-m))/(i+j), j=i..m-1), i=1..m-1), m=0..n):
%p A355503 seq(a(n), n=0..23);
%t A355503 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 *)
%t A355503 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 *)
%Y A355503 Cf. A000108, A074962, A078920, A123352, A355400.
%Y A355503 Antidiagonal sums of A368025.
%K A355503 nonn
%O A355503 0,2
%A A355503 _Alois P. Heinz_, Jul 04 2022