A131112 T(n,k) = 4*binomial(n,k) - 3*I(n,k), where I is the identity matrix; triangle T read by rows (n >= 0 and 0 <= k <= n).

1, 4, 1, 4, 8, 1, 4, 12, 12, 1, 4, 16, 24, 16, 1, 4, 20, 40, 40, 20, 1, 4, 24, 60, 80, 60, 24, 1, 4, 28, 84, 140, 140, 84, 28, 1, 4, 32, 112, 224, 280, 224, 112, 32, 1, 4, 36, 144, 336, 504, 504, 336, 144, 36, 1

Offset: 0

Author

Gary W. Adamson, Jun 15 2007

Keywords

• nonn
• tabl
• easy
• less

Examples

			Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
  1;
  4,  1;
  4,  8,  1;
  4, 12, 12,  1;
  4, 16, 24, 16,  1;
  4, 20, 40, 40, 20, 1;
  ...
		

Links

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

Crossrefs

Cf. A007318, A036563, A131110, A131111, A131113, A131114, A131115.

Programs

GAP

T:= function(n,k)
    if k=n then return 1;
    else return 4*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 4*Binomial(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Nov 18 2019

Maple

seq(seq(`if`(k=n, 1, 4*binomial(n,k)), k=0..n), n=0..10); # G. C. Greubel, Nov 18 2019

Mathematica

Table[If[k==n, 1, 4*Binomial[n, k]], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 18 2019 *)

PARI

T(n,k) = if(k==n, 1, 4*binomial(n,k)); \\ G. C. Greubel, Nov 18 2019

Sage

def T(n, k):
    if (k==n): return 1
    else: return 4*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) = 4*A007318(n,k) - 3*I(n,k), where A007318 = Pascal's triangle and I = Identity matrix.

n-th row sum = A036563(n+2) = 2^(n+2) - 3.

Bivariate o.g.f.: Sum_{n,k>=0} T(n,k)*x^n*y^k = (1 + 3*x - x*y)/((1 - x*y)*(1 - x - x*y)). - Petros Hadjicostas, Feb 20 2021