A133084 A007318 * A133080.

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, 8, 7, 56, 35, 56, 21, 8, 1, 9, 8, 84, 56, 126, 56, 36, 8, 1, 10, 9, 120, 84, 252, 126, 120, 36, 10, 1, 11, 10, 165, 120, 462, 252, 330, 120, 55, 10, 1

Offset: 1

Author

Gary W. Adamson, Sep 16 2007

Keywords

• nonn
• tabl

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

Examples

			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;
  ...
		

Links

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

Crossrefs

Cf. A133080, A003945, A133567.

Cf. A000292 (column 3 and 4), A000389 (column 5 and 6), A000580 (column 7).

Programs

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

Maple

A133084 := proc(n,k)
    add(binomial(n-1,i-1)*A133080(i,k),i=1..n) ;
end proc: # R. J. Mathar, Jun 13 2025

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 *)

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

Formula

Binomial transform of triangle A133080.