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A357824 Total number A(n,k) of k-tuples of semi-Dyck paths from (0,0) to (n,n-2*j) for j=0..floor(n/2); square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A357824 #29 Mar 25 2025 12:02:47
%S A357824 1,1,1,1,1,2,1,1,2,2,1,1,2,3,3,1,1,2,5,6,3,1,1,2,9,14,10,4,1,1,2,17,
%T A357824 36,42,20,4,1,1,2,33,98,190,132,35,5,1,1,2,65,276,882,980,429,70,5,1,
%U A357824 1,2,129,794,4150,7812,5705,1430,126,6,1,1,2,257,2316,19722,65300,78129,33040,4862,252,6
%N A357824 Total number A(n,k) of k-tuples of semi-Dyck paths from (0,0) to (n,n-2*j) for j=0..floor(n/2); square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H A357824 Alois P. Heinz, <a href="/A357824/b357824.txt">Antidiagonals n = 0..121, flattened</a>
%H A357824 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%F A357824 A(n,k) = Sum_{j=0..floor(n/2)} A008315(n,j)^k.
%F A357824 A(n,k) = Sum_{j=0..n} A120730(n,j)^k for k>=1, A(n,0) = A008619(n).
%e A357824 Square array A(n,k) begins:
%e A357824   1,  1,   1,    1,     1,       1,        1,         1, ...
%e A357824   1,  1,   1,    1,     1,       1,        1,         1, ...
%e A357824   2,  2,   2,    2,     2,       2,        2,         2, ...
%e A357824   2,  3,   5,    9,    17,      33,       65,       129, ...
%e A357824   3,  6,  14,   36,    98,     276,      794,      2316, ...
%e A357824   3, 10,  42,  190,   882,    4150,    19722,     94510, ...
%e A357824   4, 20, 132,  980,  7812,   65300,   562692,   4939220, ...
%e A357824   4, 35, 429, 5705, 78129, 1083425, 15105729, 211106945, ...
%p A357824 b:= proc(x, y) option remember; `if`(y<0 or y>x, 0,
%p A357824      `if`(x=0, 1, add(b(x-1, y+j), j=[-1, 1])))
%p A357824     end:
%p A357824 A:= (n, k)-> add(b(n, n-2*j)^k, j=0..n/2):
%p A357824 seq(seq(A(n, d-n), n=0..d), d=0..12);
%t A357824 b[x_, y_] := b[x, y] = If[y < 0 || y > x, 0, If[x == 0, 1, Sum[b[x - 1, y + j], {j, {-1, 1}}]]];
%t A357824 A[n_, k_] := Sum[b[n, n - 2*j]^k, { j, 0, n/2}];
%t A357824 Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* _Jean-François Alcover_, Oct 18 2022, after _Alois P. Heinz_ *)
%Y A357824 Columns k=0-7 give A008619, A001405, A000108, A003161, A129123, A361887, A382433, A361890.
%Y A357824 Rows n=1-5 give: A000012, A007395, A000051, A001550, A074511.
%Y A357824 Main diagonal gives A357825.
%Y A357824 Cf. A008315, A120730.
%K A357824 nonn,tabl
%O A357824 0,6
%A A357824 _Alois P. Heinz_, Oct 14 2022