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A378113 Number of n-tuples (p_1, p_2, ..., p_n) of Dyck paths of semilength n, such that each p_i is never below p_{i-1} and the upper path p_n only touches the x-axis at its endpoints.

%I A378113 #16 Nov 21 2024 10:22:20
%S A378113 1,1,2,23,880,105554,40446551,50637232553,209584899607676,
%T A378113 2881189188022646406,131778113962930341491415,
%U A378113 20065327661524165382215337625,10173706896856510992170168595911888,17178054578218938036671513200907244799852,96590987238453485101729361602126273065518820938
%N A378113 Number of n-tuples (p_1, p_2, ..., p_n) of Dyck paths of semilength n, such that each p_i is never below p_{i-1} and the upper path p_n only touches the x-axis at its endpoints.
%H A378113 Alois P. Heinz, <a href="/A378113/b378113.txt">Table of n, a(n) for n = 0..62</a>
%H A378113 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%F A378113 a(n) = A378112(n,n).
%e A378113 a(2) = 2:
%e A378113           /\       /\   /\
%e A378113    (/\/\,/  \)   (/  \,/  \) .
%e A378113 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 A378113   1        2        3        4        5 /\
%e A378113             /\         /\     /\/\     /  \
%e A378113   /\/\/\   /  \/\   /\/  \   /    \   /    \ .
%p A378113 b:= proc(n, k) option remember; `if`(n=0, 1, 2^k*mul(
%p A378113       (2*(n-i)+2*k-3)/(n+2*k-1-i), i=0..k-1)*b(n-1, k))
%p A378113     end:
%p A378113 A:= proc(n, k) option remember;
%p A378113       b(n, k)-add(A(n-i, k)*b(i, k), i=1..n-1)
%p A378113     end:
%p A378113 a:= n-> A(n$2):
%p A378113 seq(a(n), n=0..15);
%Y A378113 Main diagonal of A378112.
%Y A378113 Cf. A000108, A355400.
%K A378113 nonn
%O A378113 0,3
%A A378113 _Alois P. Heinz_, Nov 16 2024