A132046 Triangle read by rows: T(n,0) = T(n,n) = 1, and T(n,k) = 2*binomial(n,k) for 1 <= k <= n - 1.

1, 1, 1, 1, 4, 1, 1, 6, 6, 1, 1, 8, 12, 8, 1, 1, 10, 20, 20, 10, 1, 1, 12, 30, 40, 30, 12, 1, 1, 14, 42, 70, 70, 42, 14, 1, 1, 16, 56, 112, 140, 112, 56, 16, 1, 1, 18, 72, 168, 252, 252, 168, 72, 18, 1, 1, 20, 90, 240, 420, 504, 420, 240, 90, 20, 1

Offset: 0

Author

Gary W. Adamson, Aug 08 2007

Keywords

• nonn
• easy
• tabl

Comments

T(2*n,n) is A100320 (with Hankel transform A144704). - Paul Barry, Sep 19 2008

Double the internal elements of Pascal's triangle. - Paul Barry, Jan 07 2009

Coefficients of 2*(x + 1)^n - (x^n + 1) as a triangle (except for the very first term). - Thomas Baruchel, Jun 02 2018

Examples

			First few rows of the triangle are:
  1;
  1,  1;
  1,  4,  1;
  1,  6,  6,  1;
  1,  8, 12,  8,  1;
  1, 10, 20, 20, 10,  1;
  1, 12, 30, 40, 30, 12,  1;
  1, 14, 42, 70, 70, 42, 14, 1;
  ...
		

Crossrefs

Row sums: A095121.

Cf. A007318, A103451, A132046.

Cf. A154327 (diagonal sums). [Paul Barry, Jan 07 2009]

Cf. A141540.

Programs

Mathematica

T[n_, k_] := If[n == k || k == 0, 1, If[k <= n, 2 Binomial[n, k], 0]]
Flatten[Table[T[n, k], {n, 0, 20}, {k, 0, n}]] (* Emanuele Munarini, May 15 2018 *)

Maxima

T(n, k) := if k = 0 or k = n then 1 else 2*binomial(n, k)$
create_list(T(n, k), n, 0, 20, k, 0, n); /* Franck Maminirina Ramaharo, Jan 03 2019 */

Formula

T(n,k) = 2*A007318(n,k) - A103451(n,k).

T(n,k) = [k<=n] (0^(n + k) + C(n,k)*(2 - 0^(n - k) - 0^k)). - Paul Barry, Sep 19 2008

T(n,k) = A007318(n,k)*A154325(n,k). - Paul Barry, Jan 07 2009

From Emanuele Munarini, May 15 2018: (Start)

G.f.: (1 - t - x*t + 3*x*t^2 - x*t^3 - x^2*t^3)/((1 - t)*(1 - x*t)*(1 - t - x*t)).

T(n+3,k+2) = 2*T(n+2,k+2) - T(n+1,k+2) + 2*T(n+2,k+1) - 3*T(n+1,k+1) - T(n+1,k) + T(n,k+1) + T(n,k), except for n = 0 and k = 0. (End)

E.g.f.: 1 - exp(t) - exp(t*x) + 2*exp(t*(1 + x)). - Franck Maminirina Ramaharo, Jan 02 2019