A134059 Triangle T(n, k) = 3*binomial(n,k) with T(0, 0) = 1, read by rows.

1, 3, 3, 3, 6, 3, 3, 9, 9, 3, 3, 12, 18, 12, 3, 3, 15, 30, 30, 15, 3, 3, 18, 45, 60, 45, 18, 3, 3, 21, 63, 105, 105, 63, 21, 3, 3, 24, 84, 168, 210, 168, 84, 24, 3, 3, 27, 108, 252, 378, 378, 252, 108, 27, 3

Offset: 0

Author

Gary W. Adamson, Oct 05 2007

Keywords

• nonn
• tabl
• easy

Comments

Triangle T(n,k), 0 <= k <= n, read by rows given by [3, -2, 0, 0, 0, 0, 0, ...] DELTA [3, -2, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 07 2007

Examples

			First few rows of the triangle:
  1;
  3,  3;
  3,  6,  3;
  3,  9,  9,  3;
  3, 12, 18, 12,  3;
  3, 15, 30, 30, 15,  3;
  3, 18, 45, 60, 45, 18,  3;
  ...
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..5150 (rows n = 0..100, flattened)

Crossrefs

Cf. A007318, A082505 (row sums), A084938, A134058.

Programs

Magma

A134059:= func< n,k | n eq 0 select 1 else 3*Binomial(n,k) >;
[A134059(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 26 2021

Mathematica

Join[{1},Rest[Flatten[Table[3Binomial[n,k],{n,0,10},{k,0,n}]]]] (* Harvey P. Dale, Feb 15 2014 *)
Table[3*Binomial[n,k] -2*Boole[n==0], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 26 2021 *)

Sage

def A134059(n,k): return 3*binomial(n,k) - 2*bool(n==0)
flatten([[A134059(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 26 2021

Formula

3*Pascal's triangle A007318, then replace T(0,0) with 1.

G.f.: Sum_{n>=0} Sum_{k>=0} T(n,k) *x^n * y^k = 1 - 3*(1+y)*x/(x+x*y-1). - R. J. Mathar, Feb 19 2020

From G. C. Greubel, Apr 27 2021: (Start)

T(n, k) = 3*binomial(n,k) - 2*[n=0].

Sum_{k=0..n} T(n, k) = 3*2^n - 2*[n=0] = A082505(n+1). (End)

E.g.f.: 3*exp(x*(1+y)) - 2. - Stefano Spezia, Apr 03 2024

Extensions

Title changed by G. C. Greubel, Apr 26 2021