A126791 Binomial matrix applied to A111418.

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

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 from 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

From R. J. Mathar, Mar 12 2013: (Start)

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; ... (End)

Links

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

Formula

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

Sum_{k=0..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

The n-th row polynomial R(n,x) of the row-reversed triangle equals the n-th degree Taylor polynomial of the function (1 + x)*(1 + 3*x + x^2)^n expanded about the point x = 0. - Peter Bala, Sep 06 2022

From Peter Bala, Nov 16 2025: (Start)

Riordan array ( (sqrt((1 - x)/(1 - 5*x)) - 1)/(2*x), (1 - 3*x - sqrt((1 - x)*(1 - 5*x)))/(2*x) ).

Triangle T = A007318^3 * A061554 (triangle version).

The square array T * transpose(T) is the Hankel matrix corresponding to A026378 (the first column of T). (End)

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; ...
From Philippe Deléham, Nov 07 2011: (Start)
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 (End)

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
# Alternative:
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 *)

Prog

(Python)
def A126791(n, k): # for larger n, use @cache from functools
    return (A126791(n-1, k-1) + 3*A126791(n-1, k) if k else 4*A126791(n-1, 0)
        ) + A126791(n-1, k+1) if 0 <= k < n else int(k == n)
# M. F. Hasler, Nov 22 2025

Crossrefs

Cf. A007318, A026378, A061554.

Sequence in context: A369912 A093035 A301624 * A052179 A171589 A126331

Adjacent sequences: A126788 A126789 A126790 * A126792 A126793 A126794

Keyword

nonn,tabl

Author

Philippe Deléham, Mar 14 2007

Status

approved