A131112 - OEIS

%I #28 Sep 08 2022 08:45:30

%S 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,

%T 1,4,28,84,140,140,84,28,1,4,32,112,224,280,224,112,32,1,4,36,144,336,

%U 504,504,336,144,36,1

%N 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).

%H G. C. Greubel, <a href="/A131112/b131112.txt">Rows n = 0..100 of triangle, flattened</a>

%F T(n,k) = 4*A007318(n,k) - 3*I(n,k), where A007318 = Pascal's triangle and I = Identity matrix.

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

%F 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

%e Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:

%e   1;

%e   4,  1;

%e   4,  8,  1;

%e   4, 12, 12,  1;

%e   4, 16, 24, 16,  1;

%e   4, 20, 40, 40, 20, 1;

%e   ...

%p seq(seq(`if`(k=n, 1, 4*binomial(n,k)), k=0..n), n=0..10); # _G. C. Greubel_, Nov 18 2019

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

%o (PARI) T(n,k) = if(k==n, 1, 4*binomial(n,k)); \\ _G. C. Greubel_, Nov 18 2019

%o (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

%o (Sage)

%o def T(n, k):

%o     if (k==n): return 1

%o     else: return 4*binomial(n, k)

%o [[T(n, k) for k in (0..n)] for n in (0..10)]

%o # _G. C. Greubel_, Nov 18 2019

%o (GAP)

%o T:= function(n,k)

%o     if k=n then return 1;

%o     else return 4*Binomial(n,k);

%o     fi;  end;

%o Flat(List([0..10], n-> List([0..n], k-> T(n,k) ))); # _G. C. Greubel_, Nov 18 2019

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

%K nonn,tabl,easy,less

%O 0,2

%A _Gary W. Adamson_, Jun 15 2007