OFFSET
1,2
COMMENTS
Row sums = A003945: (1, 3, 6, 12, 24, 48, 96, ...).
A133084 is jointly generated with A133567 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. - Clark Kimberling, Feb 28 2012
LINKS
G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
FORMULA
Binomial transform of triangle A133080.
EXAMPLE
First few rows of the triangle:
1;
2, 1;
3, 2, 1;
4, 3, 4, 1;
5, 4, 10, 4, 1;
6, 5, 20, 10, 6, 1;
7, 6, 35, 20, 21, 6, 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 *)
T[n_, k_] := If[k == n, 1, (1 - (1 + (-1)^k)/2 )*Binomial[n, k] + ((1 + (-1)^k)/2)*Binomial[n - 1, k - 1]]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] (* G. C. Greubel, Oct 21 2017 *)
PROG
(PARI) for(n=1, 10, for(k=1, n, print1(if(k == n, 1, (1 - (1 + (-1)^k)/2 )*binomial(n, k) + ((1 + (-1)^k)/2)*binomial(n - 1, k - 1)), ", "))) \\ G. C. Greubel, Oct 21 2017
(Magma) /* As triangle */ [[(1-(1+(-1)^k)/2 )*Binomial(n, k)+((1+(-1)^k)/2)*Binomial(n-1, k-1): k in [1..n]]: n in [1.. 11]]; // Vincenzo Librandi, Oct 21 2017
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, Sep 16 2007
STATUS
approved