A378113 - OEIS

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 #16 Nov 21 2024 10:22:20

%S 1,1,2,23,880,105554,40446551,50637232553,209584899607676,

%T 2881189188022646406,131778113962930341491415,

%U 20065327661524165382215337625,10173706896856510992170168595911888,17178054578218938036671513200907244799852,96590987238453485101729361602126273065518820938

%N 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 Alois P. Heinz, <a href="/A378113/b378113.txt">Table of n, a(n) for n = 0..62</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>

%F a(n) = A378112(n,n).

%e a(2) = 2:

%e /\ /\ /\

%e (/\/\,/ \) (/ \,/ \) .

%e 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 1 2 3 4 5 /\

%e /\ /\ /\/\ / \

%e /\/\/\ / \/\ /\/ \ / \ / \ .

%p b:= proc(n, k) option remember; `if`(n=0, 1, 2^k*mul(

%p (2*(n-i)+2*k-3)/(n+2*k-1-i), i=0..k-1)*b(n-1, k))

%p end:

%p A:= proc(n, k) option remember;

%p b(n, k)-add(A(n-i, k)*b(i, k), i=1..n-1)

%p end:

%p a:= n-> A(n$2):

%p seq(a(n), n=0..15);

%Y Main diagonal of A378112.

%Y Cf. A000108, A355400.

%K nonn,new

%O 0,3

%A _Alois P. Heinz_, Nov 16 2024