This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A288972 #23 Oct 18 2018 16:02:14
%S A288972 1,1,1,1,1,1,1,1,2,1,1,1,9,10,1,1,1,44,471,92,1,1,1,225,27076,82899,
%T A288972 1348,1,1,1,1182,1713955,102695344,36913581,28808,1,1,1,6321,
%U A288972 114751470,147556480375,1565018426896,34878248649,845800,1
%N A288972 Number A(n,k) of Dyck paths having exactly k peaks in each of the levels 1,...,n and no other peaks; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%C A288972 The semilengths of Dyck paths counted by A(n,k) are elements of the integer interval [k*n+n-1, k*n*(n+1)/2] for n,k>0.
%H A288972 Alois P. Heinz, <a href="/A288972/b288972.txt">Antidiagonals n = 0..26, flattened</a>
%H A288972 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lattice_path#Counting_lattice_paths">Counting lattice paths</a>
%e A288972 . A(3,1) = 10:
%e A288972 .
%e A288972 . /\ /\ /\ /\
%e A288972 . /\/ \ / \/\ /\/ \ / \/\
%e A288972 . /\/ \ /\/ \ / \/\ / \/\
%e A288972 .
%e A288972 . /\ /\ /\
%e A288972 . /\ / \ / \ /\ /\ / \
%e A288972 . /\/ \/ \ /\/ \/ \ / \/\/ \
%e A288972 .
%e A288972 . /\ /\ /\
%e A288972 . /\ / \ / \ /\ / \ /\
%e A288972 . / \/ \/\ / \/\/ \ / \/ \/\ .
%e A288972 .
%e A288972 Square array A(n,k) begins:
%e A288972 1, 1, 1, 1, 1, ...
%e A288972 1, 1, 1, 1, 1, ...
%e A288972 1, 2, 9, 44, 225, ...
%e A288972 1, 10, 471, 27076, 1713955, ...
%e A288972 1, 92, 82899, 102695344, 147556480375, ...
%e A288972 1, 1348, 36913581, 1565018426896, 81072887990665625, ...
%p A288972 b:= proc(n, k, j, v) option remember; `if`(n=j, `if`(v=1, 1, 0),
%p A288972 `if`(v<2, 0, add(b(n-j, k, i, v-1)*(binomial(i, k)*
%p A288972 binomial(j-1, i-1-k)), i=1..min(j+k, n-j))))
%p A288972 end:
%p A288972 A:= proc(n, k) option remember; `if`(n=0 or k=0, 1,
%p A288972 add(b(w, k, k, n), w=k*n+n-1..k*n*(n+1)/2))
%p A288972 end:
%p A288972 seq(seq(A(n, d-n), n=0..d), d=0..10);
%t A288972 b[n_, k_, j_, v_]:=b[n, k, j, v]=If[n==j, If[v==1, 1, 0], If[v<2, 0, Sum[b[n - j, k, i, v - 1] Binomial[i, k] Binomial[j - 1, i - 1 - k], {i, Min[j + k, n - j]}]]]; A[n_, k_]:=A[n, k]=If[n==0 || k==0, 1, Sum[b[w, k, k, n], {w, k*n + n - 1, k*n*(n + 1)/2}]]; Table[A[n, d - n], {d, 0, 10}, {n, 0, d}] // Flatten (* _Indranil Ghosh_, Jul 06 2017, after Maple code *)
%Y A288972 Columns k=0-2 give: A000012, A289020, A289054.
%Y A288972 Rows n=0+1,2,3 give: A000012, A176479, A289030.
%Y A288972 Main diagonal gives A288940.
%K A288972 nonn,tabl
%O A288972 0,9
%A A288972 _Alois P. Heinz_, Jun 20 2017