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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).

%I A131111 #34 Sep 19 2024 01:51:52
%S A131111 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,
%T A131111 3,21,63,105,105,63,21,1,3,24,84,168,210,168,84,24,1,3,27,108,252,378,
%U A131111 378,252,108,27,1
%N 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).
%C A131111 Row sums = A033484: (1, 4, 10, 22, 46, ...) = 3*2^n - 2.
%H A131111 G. C. Greubel, <a href="/A131111/b131111.txt">Rows n = 0..100 of triangle, flattened</a>
%F A131111 T(n,k) = 3*A007318(n,k) - 2*I(n,k), where A007318 = Pascal's triangle and I = Identity matrix.
%F A131111 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
%e A131111 Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
%e A131111   1;
%e A131111   3,  1;
%e A131111   3,  6,  1;
%e A131111   3,  9,  9,  1;
%e A131111   3, 12, 18, 12,  1;
%e A131111   3, 15, 30, 30, 15,  1;
%e A131111   3, 18, 45, 60, 45, 18, 1;
%e A131111   ...
%p A131111 seq(seq(`if`(k=n, 1, 3*binomial(n,k)), k=0..n), n=0..10); # _G. C. Greubel_, Nov 18 2019
%t A131111 Table[If[k==n, 1, 3*Binomial[n, k]], {n,0,10}, {k,0,n}]//Flatten (* _G. C. Greubel_, Nov 18 2019 *)
%o A131111 (PARI) T(n,k) = if(k==n, 1, 3*binomial(n,k)); \\ _G. C. Greubel_, Nov 18 2019
%o A131111 (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
%o A131111 (Sage)
%o A131111 @CachedFunction
%o A131111 def T(n, k):
%o A131111     if (k==n): return 1
%o A131111     else: return 3*binomial(n, k)
%o A131111 [[T(n, k) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Nov 18 2019
%o A131111 (GAP)
%o A131111 T:= function(n,k)
%o A131111     if k=n then return 1;
%o A131111     else return 3*Binomial(n,k);
%o A131111     fi;  end;
%o A131111 Flat(List([0..10], n-> List([0..n], k-> T(n,k) ))); # _G. C. Greubel_, Nov 18 2019
%Y A131111 Cf. A131110, A131112, A131113, A131114, A131115.
%K A131111 nonn,tabl,easy,less
%O A131111 0,2
%A A131111 _Gary W. Adamson_, Jun 15 2007