A126953 - OEIS

%I #20 Sep 17 2024 20:55:15

%S 1,3,1,10,3,1,33,11,3,1,110,36,12,3,1,366,122,39,13,3,1,1220,405,135,

%T 42,14,3,1,4065,1355,447,149,45,15,3,1,13550,4512,1504,492,164,48,16,

%U 3,1,45162,15054,5004,1668,540,180,51,17,3,1

%N Triangle T(n,k), 0 <= k <= n, read by rows given by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = 3*T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) + T(n-1,k+1) for k >= 1.

%C This triangle belongs to the family of triangles defined by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,0) = x*T(n-1,0) + T(n-1,1), T(n,k) = T(n-1,k-1) + y*T(n-1,k) + T(n-1,k+1) for k >= 1. Other triangles arise from choosing different values for (x,y): (0,0) -> A053121; (0,1) -> A089942; (0,2) -> A126093; (0,3) -> A126970; (1,0)-> A061554; (1,1) -> A064189; (1,2) -> A039599; (1,3) -> A110877; (1,4) -> A124576; (2,0) -> A126075; (2,1) -> A038622; (2,2) -> A039598; (2,3) -> A124733; (2,4) -> A124575; (3,0) -> A126953; (3,1) -> A126954; (3,2) -> A111418; (3,3) -> A091965; (3,4) -> A124574; (4,3) -> A126791; (4,4) -> A052179; (4,5) -> A126331; (5,5) -> A125906. - _Philippe Deléham_, Sep 25 2007

%C Riordan array (2/(1-6x+sqrt(1-4*x^2)),x*c(x^2)) where c(x)= g.f. of the Catalan numbers A000108. - _Philippe Deléham_, Jun 01 2013

%H G. C. Greubel, <a href="/A126953/b126953.txt">Table of n, a(n) for the first 50 rows, flattened</a>

%F Sum_{k=0..n} T(n,k) = A127359(n).

%F Sum_{k>=0} T(m,k)*T(n,k) = T(m+n,0) = A126931(m+n).

%F Sum_{k=0..n} T(n,k)*(-2*k+1) = 2^n. - _Philippe Deléham_, Mar 25 2007

%e Triangle begins:

%e      1;

%e      3,    1;

%e     10,    3,   1;

%e     33,   11,   3,   1;

%e    110,   36,  12,   3,  1;

%e    366,  122,  39,  13,  3,  1;

%e   1220,  405, 135,  42, 14,  3, 1;

%e   4065, 1355, 447, 149, 45, 15, 3, 1;

%t T[0, 0, x_, y_] := 1; T[n_, 0, x_, y_] := x*T[n - 1, 0, x, y] + T[n - 1, 1, x, y]; T[n_, k_, x_, y_] := T[n, k, x, y] = If[k < 0 || k > n, 0, T[n - 1, k - 1, x, y] + y*T[n - 1, k, x, y] + T[n - 1, k + 1, x, y]];

%t Table[T[n, k, 3, 0], {n, 0, 10}, {k, 0, n}] // Flatten (* _G. C. Greubel_, Apr 21 2017 *)

%K nonn,tabl

%O 0,2

%A _Philippe Deléham_, Mar 19 2007