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A126791 Binomial matrix applied to A111418. 24
1, 4, 1, 17, 7, 1, 75, 39, 10, 1, 339, 202, 70, 13, 1, 1558, 1015, 425, 110, 16, 1, 7247, 5028, 2400, 771, 159, 19, 1, 34016, 24731, 12999, 4872, 1267, 217, 22, 1, 160795, 121208, 68600, 28882, 8890, 1940, 284, 25, 1, 764388, 593019, 355890, 164136 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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)+3*T(n-1,k)+T(n-1,k+1) for k>=1.

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

The matrix inverse starts

1;

-4,1;

11,-7,1;

-29,31,-10,1;

76,-115,60,-13,1;

-199,390,-285,98,-16,1;

521,-1254,1185,-566,145,-19,1;

-1364,3893,-4524,2785,-985,201,-22,1; ... - R. J. Mathar, Mar 12 2013

LINKS

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

FORMULA

Sum{k, k>=0}T(m,k)*T(n,k) = T(m+n,0) = A026378(m+n+1).

Sum{k, 0<=k<=n}T(n,k) = 5^n = A000351(n).

T(n,k) = (-1)^(n-k)*(GegenbauerC(n-k,-n+1,3/2) - GegenbauerC(n-k-1,-n+1,3/2)). - Peter Luschny, May 13 2016

EXAMPLE

Triangle begins:

1;

4, 1;

17, 7, 1;

75, 39, 10, 1;

339, 202, 70, 13, 1;

1558, 1015, 425, 110, 16, 1;

7247, 5028, 2400, 771, 159, 19, 1;

34016, 24731, 12999, 4872, 1267, 217, 22, 1;...

Production matrix begins

4, 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

- From Philippe Deléham, Nov 07 2011

MAPLE

A126791 := proc(n, k)

    if n=0 and k = 0 then

        1 ;

    elif k <0 or k>n then

        0;

    elif k= 0 then

        4*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, Mar 12 2013

T := (n, k) -> (-1)^(n-k)*simplify(GegenbauerC(n-k, -n+1, 3/2) - GegenbauerC(n-k-1, -n+1, 3/2)): seq(seq(T(n, k), k=1..n), n=1..10); # Peter Luschny, May 13 2016

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

CROSSREFS

Sequence in context: A209411 A093035 A301624 * A052179 A171589 A126331

Adjacent sequences:  A126788 A126789 A126790 * A126792 A126793 A126794

KEYWORD

nonn,tabl

AUTHOR

Philippe Deléham, Mar 14 2007

STATUS

approved

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Last modified October 20 00:45 EDT 2018. Contains 316378 sequences. (Running on oeis4.)