cp's OEIS Frontend

A378112 Number A(n,k) of k-tuples (p_1, p_2, ..., p_k) of Dyck paths of semilength n, such that each p_i is never below p_{i-1} and the upper path p_k only touches the x-axis at its endpoints; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A378112 #24 Nov 16 2024 19:30:34
%S A378112 1,1,1,1,1,0,1,1,1,0,1,1,2,2,0,1,1,3,9,5,0,1,1,4,23,55,14,0,1,1,5,46,
%T A378112 265,400,42,0,1,1,6,80,880,3942,3266,132,0,1,1,7,127,2347,23695,70395,
%U A378112 28999,429,0,1,1,8,189,5403,105554,824229,1445700,274537,1430,0
%N A378112 Number A(n,k) of k-tuples (p_1, p_2, ..., p_k) of Dyck paths of semilength n, such that each p_i is never below p_{i-1} and the upper path p_k only touches the x-axis at its endpoints; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H A378112 Alois P. Heinz, <a href="/A378112/b378112.txt">Antidiagonals n = 0..115, flattened</a>
%H A378112 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%F A378112 Column k is INVERTi transform of row k of A368025.
%e A378112 A(3,2) = 9:
%e A378112                              /\
%e A378112            /\/\             /  \      /\     /\/\
%e A378112   (/\/\/\,/    \)  (/\/\/\,/    \)  (/  \/\,/    \)
%e A378112 .
%e A378112             /\                                /\
%e A378112     /\     /  \        /\   /\/\        /\   /  \
%e A378112   (/  \/\,/    \)  (/\/  \,/    \)  (/\/  \,/    \)
%e A378112 .
%e A378112                              /\        /\     /\
%e A378112     /\/\   /\/\      /\/\   /  \      /  \   /  \
%e A378112   (/    \,/    \)  (/    \,/    \)  (/    \,/    \)
%e A378112 .
%e A378112 Square array A(n,k) begins:
%e A378112   1,  1,    1,     1,      1,       1,        1, ...
%e A378112   1,  1,    1,     1,      1,       1,        1, ...
%e A378112   0,  1,    2,     3,      4,       5,        6, ...
%e A378112   0,  2,    9,    23,     46,      80,      127, ...
%e A378112   0,  5,   55,   265,    880,    2347,     5403, ...
%e A378112   0, 14,  400,  3942,  23695,  105554,   382508, ...
%e A378112   0, 42, 3266, 70395, 824229, 6601728, 40446551, ...
%p A378112 b:= proc(n, k) option remember; `if`(n=0, 1, 2^k*mul(
%p A378112       (2*(n-i)+2*k-3)/(n+2*k-1-i), i=0..k-1)*b(n-1, k))
%p A378112     end:
%p A378112 A:= proc(n, k) option remember;
%p A378112       b(n, k)-add(A(n-i, k)*b(i, k), i=1..n-1)
%p A378112     end:
%p A378112 seq(seq(A(n, d-n), n=0..d), d=0..10);
%Y A378112 Columns k=0-3 give: A019590(n+1), A120588, A355281, A378114.
%Y A378112 Rows n=0+1,2,3 give: A000012, A001477, A101986.
%Y A378112 Main diagonal gives A378113.
%Y A378112 Cf. A000108, A078920, A123352, A368025.
%K A378112 nonn,tabl
%O A378112 0,13
%A A378112 _Alois P. Heinz_, Nov 16 2024