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 A336858 #25 Nov 29 2023 08:49:25
%S A336858 1,1,1,1,3,1,1,5,9,1,1,7,21,31,1,1,9,37,89,121,1,1,11,57,183,393,515,
%T A336858 1,1,13,81,321,897,1805,2321,1,1,15,109,511,1729,4431,8557,10879,1,1,
%U A336858 17,141,761,3001,9161,22149,41585,52465,1,1,19,177,1079,4841,17003,48313,112047,206097,258563,1
%N A336858 Triangle read by rows: T(n,k) = T(n,k-1) + T(n-1, k) + T(n-1,k-1) with T(n,0) = T(n, n) = 1 (n >= 0, 0 <= k <= n).
%C A336858 This is _J. M. Bergot_'s triangular array described in A104858 with the top vertex of the triangle shifted from (1,1) to (0,0).
%F A336858 T(n,k) = T(n, k-1) + T(n-1, k) + T(n-1, k-1) for 1 <= k <= n-1 with T(n,0) = 1 = T(n,n) for n >= 0.
%F A336858 T(n,k) = D(n,k) - Sum_{m=1..k} b(m-1)*D(n-m, k-m) - Sum_{m=0..k-1} D(n-m, k-m-1), where D(n,k) = A008288(n,k) (square array of Delannoy numbers) and b(n) = A086616(n).
%F A336858 T(n,1) = A005408(n-1) = 2*n - 1 for n >= 1.
%F A336858 T(n,2) = A059993(n-2) = 2*n^2 - 2*n - 3 for n >= 2.
%F A336858 T(n,n-1) = A086616(n-1) for n >= 1.
%F A336858 T(n,n-2) = A035011(n-1) = A006318(n-1) - 1 for n >= 2.
%F A336858 Sum_{k=0..n} T(n,k) = A104858(n) for n >= 0.
%F A336858 Bivariate o.g.f.: (1 - y - x*y*(1 + g(x*y)))/((1 - x*y)*(1 - x - y - x*y)), where g(w) = 2/(1 - w + sqrt(1 - 6*w + w^2)) = o.g.f. of A006318 (large Schroeder numbers).
%F A336858 Bivariate o.g.f.: (1 - y - 2*x*y*q(x*y))/((1 - x*y)*(1 - x - y - x*y)), where q(w) = 2/(1 + w + sqrt(1 - 6*w + w^2)) = o.g.f. of A001003 (little Schroeder numbers).
%F A336858 T(2*n,n) = A333090(n). - _Peter Luschny_, Aug 06 2020
%e A336858 Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
%e A336858 1;
%e A336858 1, 1;
%e A336858 1, 3, 1;
%e A336858 1, 5, 9, 1;
%e A336858 1, 7, 21, 31, 1;
%e A336858 1, 9, 37, 89, 121, 1;
%e A336858 1, 11, 57, 183, 393, 515, 1;
%e A336858 1, 13, 81, 321, 897, 1805, 2321, 1;
%e A336858 1, 15, 109, 511, 1729, 4431, 8557, 10879, 1;
%e A336858 ...
%p A336858 A336858row := proc(n) option remember; local T, k, row;
%p A336858 row := Array(0..n, fill=1);
%p A336858 if n = 0 then return row fi; T := procname(n-1);
%p A336858 for k from 1 to n-1 do row[k] := T[k] + T[k-1] + row[k-1] od; row end:
%p A336858 T := (n, k) -> A336858row(n)[k]:
%p A336858 seq(print(seq(T(n, k), k=0..n)), n=0..8); # _Peter Luschny_, Aug 06 2020
%t A336858 T[_, 0] = 1; T[n_, n_] = 1;
%t A336858 T[n_, k_] := T[n, k] = T[n, k-1] + T[n-1, k] + T[n-1, k-1];
%t A336858 Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Nov 29 2023 *)
%Y A336858 Cf. A001003, A005408, A006318, A008288, A035011, A059993, A086616, A104858, A333090.
%K A336858 nonn,tabl
%O A336858 0,5
%A A336858 _Petros Hadjicostas_, Aug 05 2020