A336858 - OEIS

%I #25 Nov 29 2023 08:49:25

%S 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 1,1,13,81,321,897,1805,2321,1,1,15,109,511,1729,4431,8557,10879,1,1,

%U 17,141,761,3001,9161,22149,41585,52465,1,1,19,177,1079,4841,17003,48313,112047,206097,258563,1

%N 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 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 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 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 T(n,1) = A005408(n-1) = 2*n - 1 for n >= 1.

%F T(n,2) = A059993(n-2) = 2*n^2 - 2*n - 3 for n >= 2.

%F T(n,n-1) = A086616(n-1) for n >= 1.

%F T(n,n-2) = A035011(n-1) = A006318(n-1) - 1 for n >= 2.

%F Sum_{k=0..n} T(n,k) = A104858(n) for n >= 0.

%F 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 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 T(2*n,n) = A333090(n). - _Peter Luschny_, Aug 06 2020

%e Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:

%e   1;

%e   1,  1;

%e   1,  3,   1;

%e   1,  5,   9,   1;

%e   1,  7,  21,  31,    1;

%e   1,  9,  37,  89,  121,    1;

%e   1, 11,  57, 183,  393,  515,    1;

%e   1, 13,  81, 321,  897, 1805, 2321,     1;

%e   1, 15, 109, 511, 1729, 4431, 8557, 10879, 1;

%e   ...

%p A336858row := proc(n) option remember; local T, k, row;

%p row := Array(0..n, fill=1);

%p if n = 0 then return row fi; T := procname(n-1);

%p for k from 1 to n-1 do row[k] := T[k] + T[k-1] + row[k-1] od; row end:

%p T := (n, k) -> A336858row(n)[k]:

%p seq(print(seq(T(n, k), k=0..n)), n=0..8); # _Peter Luschny_, Aug 06 2020

%t T[_, 0] = 1; T[n_, n_] = 1;

%t T[n_, k_] := T[n, k] = T[n, k-1] + T[n-1, k] + T[n-1, k-1];

%t Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Nov 29 2023 *)

%Y Cf. A001003, A005408, A006318, A008288, A035011, A059993, A086616, A104858, A333090.

%K nonn,tabl

%O 0,5

%A _Petros Hadjicostas_, Aug 05 2020