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

0,2

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

Links

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

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

Example

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)

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 *)

Crossrefs

Cf. A182401.

Sequence in context: A075500 A096645 A140713 * A146414 A146374 A188647

Adjacent sequences: A125903 A125904 A125905 * A125907 A125908 A125909

Keyword

nonn,tabl

Author

Philippe Deléham, Feb 04 2007

Status

approved