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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).

%I A131112 #28 Sep 08 2022 08:45:30
%S A131112 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 A131112 1,4,28,84,140,140,84,28,1,4,32,112,224,280,224,112,32,1,4,36,144,336,
%U A131112 504,504,336,144,36,1
%N 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).
%H A131112 G. C. Greubel, <a href="/A131112/b131112.txt">Rows n = 0..100 of triangle, flattened</a>
%F A131112 T(n,k) = 4*A007318(n,k) - 3*I(n,k), where A007318 = Pascal's triangle and I = Identity matrix.
%F A131112 n-th row sum = A036563(n+2) = 2^(n+2) - 3.
%F A131112 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 A131112 Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
%e A131112   1;
%e A131112   4,  1;
%e A131112   4,  8,  1;
%e A131112   4, 12, 12,  1;
%e A131112   4, 16, 24, 16,  1;
%e A131112   4, 20, 40, 40, 20, 1;
%e A131112   ...
%p A131112 seq(seq(`if`(k=n, 1, 4*binomial(n,k)), k=0..n), n=0..10); # _G. C. Greubel_, Nov 18 2019
%t A131112 Table[If[k==n, 1, 4*Binomial[n, k]], {n,0,10}, {k,0,n}]//Flatten (* _G. C. Greubel_, Nov 18 2019 *)
%o A131112 (PARI) T(n,k) = if(k==n, 1, 4*binomial(n,k)); \\ _G. C. Greubel_, Nov 18 2019
%o A131112 (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 A131112 (Sage)
%o A131112 def T(n, k):
%o A131112     if (k==n): return 1
%o A131112     else: return 4*binomial(n, k)
%o A131112 [[T(n, k) for k in (0..n)] for n in (0..10)]
%o A131112 # _G. C. Greubel_, Nov 18 2019
%o A131112 (GAP)
%o A131112 T:= function(n,k)
%o A131112     if k=n then return 1;
%o A131112     else return 4*Binomial(n,k);
%o A131112     fi;  end;
%o A131112 Flat(List([0..10], n-> List([0..n], k-> T(n,k) ))); # _G. C. Greubel_, Nov 18 2019
%Y A131112 Cf. A007318, A036563, A131110, A131111, A131113, A131114, A131115.
%K A131112 nonn,tabl,easy,less
%O A131112 0,2
%A A131112 _Gary W. Adamson_, Jun 15 2007