A126093 Inverse binomial matrix applied to A110877.

1, 0, 1, 1, 2, 1, 2, 6, 4, 1, 6, 18, 15, 6, 1, 18, 57, 54, 28, 8, 1, 57, 186, 193, 118, 45, 10, 1, 186, 622, 690, 474, 218, 66, 12, 1, 622, 2120, 2476, 1856, 976, 362, 91, 14, 1, 2120, 7338, 8928, 7164, 4170, 1791, 558, 120, 16, 1

Offset

0,5

Comments

Diagonal sums are A065601. - Philippe Deléham, Mar 05 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 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

Links

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

Yidong Sun and Luping Ma, Minors of a class of Riordan arrays related to weighted partial Motzkin paths, Eur. J. Comb. 39, 157-169 (2014), Table 2.2.

Formula

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) = T(n-1,1), T(n,k) = T(n-1,k-1) + 2*T(n-1,k) + T(n-1,k+1) for k>=1.

Sum_{k=0..n} T(m,k)*T(n,k) = T(m+n,0) = A000957(m+n+1).

Sum_{k=0..n-1} T(n,k) = A026641(n), for n>=1. - Philippe Deléham, Mar 05 2007

Sum_{k=0..n} T(n,k)*(3k+1) = 4^n. - Philippe Deléham, Mar 22 2007

Example

Triangle begins:
     1;
     0,    1;
     1,    2,    1;
     2,    6,    4,    1;
     6,   18,   15,    6,    1;
    18,   57,   54,   28,    8,    1;
    57,  186,  193,  118,   45,   10,   1;
   186,  622,  690,  474,  218,   66,  12,   1;
   622, 2120, 2476, 1856,  976,  362,  91,  14,  1;
  2120, 7338, 8928, 7164, 4170, 1791, 558, 120, 16, 1;
Production matrix begins
  0, 1;
  1, 2, 1;
  0, 1, 2, 1;
  0, 0, 1, 2, 1;
  0, 0, 0, 1, 2, 1;
  0, 0, 0, 0, 1, 2, 1;
  0, 0, 0, 0, 0, 1, 2, 1;
  0, 0, 0, 0, 0, 0, 1, 2, 1;
  0, 0, 0, 0, 0, 0, 0, 1, 2, 1;
- Philippe Deléham, Nov 07 2011

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, 0, 2], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Apr 21 2017 *)

Prog

(Sage)
@CachedFunction
def T(n, k, x, y):
    if (k<0 or k>n): return 0
    elif (n==0 and k==0): return 1
    elif (k==0): return x*T(n-1, 0, x, y) + T(n-1, 1, x, y)
    else: return T(n-1, k-1, x, y) + y*T(n-1, k, x, y) + T(n-1, k+1, x, y)
[[T(n, k, 0, 2) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jan 27 2020

Crossrefs

Sequence in context: A121341 A241737 A174959 * A065279 A343383 A151962

Adjacent sequences: A126090 A126091 A126092 * A126094 A126095 A126096

Keyword

nonn,tabl

Author

Philippe Deléham, Mar 03 2007

Status

approved