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A378114 Number of 3-tuples (p_1, p_2, p_3) of Dyck paths of semilength n, such that each p_i is never below p_{i-1} and the upper path p_3 only touches the x-axis at its endpoints.

%I A378114 #14 Nov 21 2024 10:19:50
%S A378114 1,1,3,23,265,3942,70395,1445700,33188889,834702890,22656163450,
%T A378114 656075013591,20085981787831,645418018740113,21637970282382744,
%U A378114 753157297564682541,27105935164769925549,1005184072184843625837,38295251586474334236780,1495061191885030011433707
%N A378114 Number of 3-tuples (p_1, p_2, p_3) of Dyck paths of semilength n, such that each p_i is never below p_{i-1} and the upper path p_3 only touches the x-axis at its endpoints.
%H A378114 Alois P. Heinz, <a href="/A378114/b378114.txt">Table of n, a(n) for n = 0..567</a>
%H A378114 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%F A378114 INVERTi transform of A006149.
%e A378114 a(2) = 3:
%e A378114                /\            /\   /\       /\   /\   /\
%e A378114    (/\/\,/\/\,/  \)   (/\/\,/  \,/  \)   (/  \,/  \,/  \)  .
%e A378114 The a(3) = 23 3-tuples can be encoded as 114, 115, 124, 125, 134, 135, 144, 145, 155, 224, 225, 244, 245, 255, 334, 335, 344, 345, 355, 444, 445, 455, 555, where the digits represent the following Dyck paths:
%e A378114   1        2        3        4        5 /\
%e A378114             /\         /\     /\/\     /  \
%e A378114   /\/\/\   /  \/\   /\/  \   /    \   /    \ .
%p A378114 b:= proc(n, k) option remember; `if`(n=0, 1, 2^k*mul(
%p A378114       (2*(n-i)+2*k-3)/(n+2*k-1-i), i=0..k-1)*b(n-1, k))
%p A378114     end:
%p A378114 A:= proc(n, k) option remember;
%p A378114       b(n, k)-add(A(n-i, k)*b(i, k), i=1..n-1)
%p A378114     end:
%p A378114 a:= n-> A(n, 3):
%p A378114 seq(a(n), n=0..20);
%Y A378114 Column k=3 of A378112.
%Y A378114 Cf. A000108, A006149.
%K A378114 nonn
%O A378114 0,3
%A A378114 _Alois P. Heinz_, Nov 16 2024