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A126953 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. 24
1, 3, 1, 10, 3, 1, 33, 11, 3, 1, 110, 36, 12, 3, 1, 366, 122, 39, 13, 3, 1, 1220, 405, 135, 42, 14, 3, 1, 4065, 1355, 447, 149, 45, 15, 3, 1, 13550, 4512, 1504, 492, 164, 48, 16, 3, 1, 45162, 15054, 5004, 1668, 540, 180, 51, 17, 3, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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 by 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

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

LINKS

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

FORMULA

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

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

Sum_{k, 0<=k<=n} T(n,k)*(-2*k+1) = 2^n. - Philippe Deléham, Mar 25 2007

EXAMPLE

Triangle begins:

1;

3, 1;

10, 3, 1;

33, 11, 3, 1;

110, 36, 12, 3, 1;

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

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

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

MATHEMATICA

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]];

Table[T[n, k, 3, 0], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Apr 21 2017 *)

CROSSREFS

Sequence in context: A102430 A135573 A257254 * A134284 A134285 A141811

Adjacent sequences:  A126950 A126951 A126952 * A126954 A126955 A126956

KEYWORD

nonn,tabl

AUTHOR

Philippe Deléham, Mar 19 2007

STATUS

approved

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Last modified February 21 14:40 EST 2018. Contains 299414 sequences. (Running on oeis4.)