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A126331 Triangle T(n,k),0<=k<=n, read by rows defined by :T(0,0)=1, T(n,k)=0 if k<0 or if k>n, T(n,0)=4*T(n-1,0)+T(n-1,1), T(n,k)=T(n-1,k-1)+5*T(n-1,k)+T(n-1,k+1) for k>=1. 25
1, 4, 1, 17, 9, 1, 77, 63, 14, 1, 371, 406, 134, 19, 1, 1890, 2535, 1095, 230, 24, 1, 10095, 15660, 8240, 2269, 351, 29, 1, 56040, 96635, 59129, 19936, 4053, 497, 34, 1, 320795, 598344, 412216, 162862, 40698, 6572, 668, 39, 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

7^n = (n-th row terms) dot (first n+1 odd integers). Example: 7^3 = 343 = (77, 63, 14, 1) dot (1, 3, 5, 7) = (77 + 189 + 70 + 7) = 243. - Gary W. Adamson, Jun 15 2011

LINKS

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

FORMULA

Sum_{k, 0<=k<=n}T(n,k)=A098409(n).

Sum_{k, k>=0}T(m,k)*T(n,k)=T(m+n,0)=A104455(m+n).

Sum_{k, 0<=k<=n}T(n,k)*(2*k+1)=7^n . - Philippe Deléham, Mar 26 2007

EXAMPLE

Triangle begins:

1;

4, 1;

17, 9, 1;

77, 63, 14, 1;

371, 406, 134, 19, 1;

1890, 2535, 1095, 230, 24, 1;

10095, 15660, 8240, 2269, 351, 29, 1;

Production matrix begins

4, 1

1, 5, 1

0, 1, 5, 1

0, 0, 1, 5, 1

0, 0, 0, 1, 5, 1,

0, 0, 0, 0, 1, 5, 1

0, 0, 0, 0, 0, 1, 5, 1

0, 0, 0, 0, 0, 0, 1, 5, 1

0, 0, 0, 0, 0, 0, 0, 1, 5, 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, 4, 5], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, May 22 2017 *)

CROSSREFS

Sequence in context: A126791 A052179 A171589 * A013631 A113355 A201201

Adjacent sequences:  A126328 A126329 A126330 * A126332 A126333 A126334

KEYWORD

nonn,tabl

AUTHOR

Philippe Deléham, Mar 10 2007

STATUS

approved

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Last modified October 23 19:16 EDT 2018. Contains 316530 sequences. (Running on oeis4.)