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A133567
4
1, 1, 1, 1, 3, 1, 1, 6, 3, 1, 1, 10, 6, 5, 1, 1, 15, 10, 15, 5, 1, 1, 21, 15, 35, 15, 7, 1, 1, 28, 21, 70, 35, 28, 7, 1, 1, 36, 28, 126, 70, 84, 28, 9, 1, 1, 45, 36, 210, 126, 210, 84, 45, 9, 1
OFFSET
1,5
COMMENTS
Row sums = A083329: (1, 2, 5, 11, 23, 47, 95, ...).
From Clark Kimberling, Feb 28 2012: (Start)
A133567 is jointly generated with A133084 as an array of coefficients of polynomials v(n,x): initially, u(1,x) = v(1,x) = 1; for n > 1, u(n,x) = u(n-1,x) + (x+1)*v(n-1)x and v(n,x) = x*u(n-1,x) + v(n-1,x) + 1. See the Mathematica section. (End)
FORMULA
Binomial transform of triangle A133566.
EXAMPLE
First few rows of the triangle:
1;
1, 1;
1, 3, 1;
1, 6, 3, 1;
1, 10, 6, 5, 1;
1, 15, 10, 15, 5, 1;
1, 21, 15, 35, 15, 7, 1;
...
MATHEMATICA
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := x*u[n - 1, x] + v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%] (* A133567 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A133084 *)
(* Clark Kimberling, Feb 28 2012 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, Sep 16 2007
STATUS
approved