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A126970 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)=T(n-1,1), T(n,k)=T(n-1,k-1)+3*T(n-1,k)+T(n-1,k+1) for k>=1. 26
1, 0, 1, 1, 3, 1, 3, 11, 6, 1, 11, 42, 30, 9, 1, 42, 167, 141, 58, 12, 1, 167, 684, 648, 327, 95, 15, 1, 684, 2867, 2955, 1724, 627, 141, 18, 1, 2867, 12240, 13456, 8754, 3746, 1068, 196, 21, 1, 12240, 53043, 61362, 43464, 21060, 7146, 1677, 260, 24, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

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

LINKS

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

FORMULA

Sum_{k, 0<=k<=n} T(n,k) = A126952(n).

Sum_{k, k>=0} T(m,k)*T(n,k) = T(m+n,0) = A117641(m+n).

Sum_{k, 0<=k<=n} T(n,k)*(4*k+1) = 5^n. - Philippe Deléham, Mar 22 2007

EXAMPLE

Triangle begins:

1;

0, 1;

1, 3, 1;

3, 11, 6, 1;

11, 42, 30, 9, 1;

42, 167, 141, 58, 12, 1;

167, 684, 648, 327, 95, 15, 1;...

Production matrix begins

0, 1

1, 3, 1

0, 1, 3, 1

0, 0, 1, 3, 1

0, 0, 0, 1, 3, 1

0, 0, 0, 0, 1, 3, 1

0, 0, 0, 0, 0, 1, 3, 1

0, 0, 0, 0, 0, 0, 1, 3, 1

0, 0, 0, 0, 0, 0, 0, 1, 3, 1

- Philippe Deléham, Nov 07 2011

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, 0, 3], {n, 0, 49}, {k, 0, n}] // Flatten (* G. C. Greubel, Apr 21 2017 *)

CROSSREFS

Sequence in context: A170845 A025238 A289066 * A204134 A233168 A001351

Adjacent sequences:  A126967 A126968 A126969 * A126971 A126972 A126973

KEYWORD

nonn,tabl

AUTHOR

Philippe Deléham, Mar 19 2007

STATUS

approved

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Last modified February 18 20:32 EST 2018. Contains 299330 sequences. (Running on oeis4.)