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A110877 Triangle T(n,k), 0<=k<=n, read by rows, defined by: T(0,0) = 1, T(n,k) = 0 if n<k, T(n,0) = T(n-1,0) + T(n-1,1) and for k>=1 : T(n,k) = T(n-1,k-1) + x*T(n-1,k) + T(n-1,k+1) with x = 3. 28
1, 1, 1, 2, 4, 1, 6, 15, 7, 1, 21, 58, 37, 10, 1, 79, 232, 179, 68, 13, 1, 311, 954, 837, 396, 108, 16, 1, 1265, 4010, 3861, 2133, 736, 157, 19, 1, 5275, 17156, 17726, 10996, 4498, 1226, 215, 22, 1, 22431, 74469, 81330, 55212, 25716, 8391, 1893 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Similar to A064189 (x = 1) and to A039599 (x = 2).

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

Row sums yield A126568. - Philippe Deléham, Oct 10 2007

5^n = (n-th row terms) dot (first n+1 terms in the series (1, 4, 7, 10,...)). Example for row 4: 5^4 = 625 = (21, 58, 37, 10, 1) dot (1, 4, 7, 10, 13) = (21 + 232 + 259 + 100 + 13). - Gary W. Adamson, Jun 15 2011

Riordan array (2/(1+x+sqrt(1-6*x+5*x^2)), (1-3*x-sqrt(1-6*x+5*x^2))/(2*x)). - Philippe Deléham, Mar 04 2013

LINKS

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

FORMULA

T(n, 0) = A033321(n) and for k>=1: T(n, k) = Sum_{j, j>=1} T(n-j, k-1)*A002212(j).

Sum_{k, 0<=k<=n} T(m, k)*T(n, k) = T(m+n, 0) = A033321(m+n).

The triangle may also be generated from M^n * [1,0,0,0...], where M = an infinite tridiagonal matrix with 1's in the super and subdiagonals and (1,3,3,3...) in the main diagonal. - Gary W. Adamson, Dec 17 2006

Sum_{k, 0<=k<=n} T(n,k)*(3*k+1) = 5^n. - Philippe Deléham, Feb 26 2007

Sum_{k, 0<=k<=n} T(n,k) = A126568(n). - Philippe Deléham, Oct 10 2007

EXAMPLE

Triangle begins:

1;

1, 1;

2, 4, 1;

6, 15, 7, 1;

21, 58, 37, 10, 1;

79, 232, 179, 68, 13, 1;

311, 954, 837, 396, 108, 16, 1;

1265, 4010, 3861, 2133, 736, 157, 19, 1;

5275, 17156, 17726, 10996, 4498, 1226, 215, 22, 1;

22431, 74469, 81330, 55212, 25716, 8391, 1893, 282, 25, 1;

...

Production matrix begins

1, 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

MAPLE

A110877 := proc(n, k)

    if k > n then

        0;

    elif n= 0 then

        1;

    elif k = 0 then

        procname(n-1, 0)+procname(n-1, 1) ;

    else

        procname(n-1, k-1)+3*procname(n-1, k)+procname(n-1, k+1) ;

    end if;

end proc: # R. J. Mathar, Sep 06 2013

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, 1, 3], {n, 0, 49}, {k, 0, n}] // Flatten (* G. C. Greubel, Apr 21 2017 *)

CROSSREFS

Cf. A002212, A033321, A039599, A064189.

The inverse of A126126.

Sequence in context: A105357 A167546 A011369 * A204115 A204130 A204024

Adjacent sequences:  A110874 A110875 A110876 * A110878 A110879 A110880

KEYWORD

nonn,easy,tabl

AUTHOR

Philippe Deléham, Sep 19 2005

STATUS

approved

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Last modified October 23 22:04 EDT 2018. Contains 316541 sequences. (Running on oeis4.)