A125906 Riordan array (1/(1 + 5*x + x^2), x/(1 + 5*x + x^2))^(-1); inverse of Riordan array A123967.

1, 5, 1, 26, 10, 1, 140, 77, 15, 1, 777, 540, 153, 20, 1, 4425, 3630, 1325, 254, 25, 1, 25755, 23900, 10509, 2620, 380, 30, 1, 152675, 155764, 79065, 23989, 4550, 531, 35, 1, 919139, 1010560, 575078, 203560, 47270, 7240, 707, 40, 1

Offset: 0

Author

Philippe Deléham, Feb 04 2007

Keywords

• nonn
• tabl

Comments

T(0)=A053121, T(1)=A064189, T(2)=A039598, T(3)=A091965, T(4)=A052179.

Triangle read by rows: T(n,k) = number of lattice paths from (0,0) to (n,k) that do not go below the line y=0 and consist of steps U=(1,1), D=(1,-1) and five types of steps H=(1,0); example: T(3,1)=77 because we have UDU, UUD, 25 HHU paths, 25 HUH paths and 25 UHH paths. - Philippe Deléham, Sep 25 2007

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

7^n = (n-th row terms) dot (first n+1 terms in 1,2,3,...). Example: 7^3 = 343 = (140, 77, 15, 1) dot (1, 2, 3, 4) = (140 + 154 + 45 + 4) = 343. - Gary W. Adamson, Jun 17 2011

A subset of the "family of triangles" (Deleham comment of Sep 25 2007) is the succession of binomial transforms beginning with triangle A053121, (0,0); giving -> A064189, (1,1); -> A039598, (2,2); -> A091965, (3,3); -> A052179, (4,4); -> A125906, (5,5) ->, etc; generally the binomial transform of the triangle generated from (n,n) = that generated from ((n+1),(n+1)). - Gary W. Adamson, Aug 03 2011

Riordan array (f(x), x*f(x)) where f(x) is the o.g.f. of A182401. - Philippe Deléham, Mar 04 2013

Examples

			Triangle begins
       1;
       5,       1;
      26,      10,      1;
     140,      77,     15,      1;
     777,     540,    153,     20,     1;
    4425,    3630,   1325,    254,    25,    1;
   25755,   23900,  10509,   2620,   380,   30,   1;
  152675,  155764,  79065,  23989,  4550,  531,  35,  1;
  919139, 1010560, 575078, 203560, 47270, 7240, 707, 40, 1;
From _Philippe Deléham_, Nov 07 2011: (Start)
Production matrix begins
  5, 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; (End)
		

Links

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

Crossrefs

Cf. A182401.

Programs

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

Formula

Triangle T(5) where T(x) is defined by: T(0,0)=1, T(n,k)=0 if k < 0 or if k > n, T(n,k) = T(n-1,k-1) + x*T(n-1,k) + T(n-1,k+1). Sum_{k=0..n} T(m,k)*T(n,k) = T(m+n,0). Sum_{k=0..n} T(n,k) = A122898(n).

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

T(n,0) = A182401(n). - Philippe Deléham, Mar 04 2013

The n-th row polynomial R(n,x) equals the n-th degree Taylor polynomial of the function (1 - x^2)*(1 + 5*x + x^2)^n expanded about the point x = 0. - Peter Bala, Sep 06 2022