A131111 T(n, k) = 3*binomial(n,k) - 2*I(n,k), where I is the identity matrix; triangle T read by rows (n >= 0 and 0 <= k <= n).

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

Offset: 0

Author

Gary W. Adamson, Jun 15 2007

Keywords

• nonn
• tabl
• easy
• less

Comments

Row sums = A033484: (1, 4, 10, 22, 46, ...) = 3*2^n - 2.

Examples

			Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
  1;
  3,  1;
  3,  6,  1;
  3,  9,  9,  1;
  3, 12, 18, 12,  1;
  3, 15, 30, 30, 15,  1;
  3, 18, 45, 60, 45, 18, 1;
  ...
		

Links

• G. C. Greubel, Rows n = 0..100 of triangle, flattened

Crossrefs

Cf. A131110, A131112, A131113, A131114, A131115.

Programs

GAP

T:= function(n,k)
    if k=n then return 1;
    else return 3*Binomial(n,k);
    fi;  end;
Flat(List([0..10], n-> List([0..n], k-> T(n,k) ))); # G. C. Greubel, Nov 18 2019

Magma

[k eq n select 1 else 3*Binomial(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Nov 18 2019

Maple

seq(seq(`if`(k=n, 1, 3*binomial(n,k)), k=0..n), n=0..10); # G. C. Greubel, Nov 18 2019

Mathematica

Table[If[k==n, 1, 3*Binomial[n, k]], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 18 2019 *)

PARI

T(n,k) = if(k==n, 1, 3*binomial(n,k)); \\ G. C. Greubel, Nov 18 2019

Sage

@CachedFunction
def T(n, k):
    if (k==n): return 1
    else: return 3*binomial(n, k)
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Nov 18 2019

Formula

T(n,k) = 3*A007318(n,k) - 2*I(n,k), where A007318 = Pascal's triangle and I = Identity matrix.

Bivariate o.g.f.: Sum_{n,k>=0} T(n,k)*x^n*y^k = (1 + 2*x - x*y)/((1 - x*y)*(1 - x - x*y)). - Petros Hadjicostas, Feb 20 2021