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A131114 T(n,k) = 6*binomial(n,k) - 5*I(n,k), where I is the identity matrix; triangle T read by rows (n >= 0 and 0 <= k <= n).

%I A131114 #27 Sep 08 2022 08:45:30
%S A131114 1,6,1,6,12,1,6,18,18,1,6,24,36,24,1,6,30,60,60,30,1,6,36,90,120,90,
%T A131114 36,1,6,42,126,210,210,126,42,1,6,48,168,336,420,336,168,48,1,6,54,
%U A131114 216,504,756,756,504,216,54,1
%N A131114 T(n,k) = 6*binomial(n,k) - 5*I(n,k), where I is the identity matrix; triangle T read by rows (n >= 0 and 0 <= k <= n).
%C A131114 Row sums give A048488.
%H A131114 G. C. Greubel, <a href="/A131114/b131114.txt">Rows n = 0..100 of triangle, flattened</a>
%F A131114 T(n,k) = 6*A007318(n,k) - 5*I(n,k), where A007318 = Pascal's triangle and I = Identity matrix.
%F A131114 Bivariate o.g.f.: Sum_{n,k>=0} T(n,k)*x^n*y^k = (1 + 5*x - x*y)/((1 - x*y)*(1 - x - x*y)).
%e A131114 Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
%e A131114   1;
%e A131114   6,  1;
%e A131114   6, 12,  1;
%e A131114   6, 18, 18,   1;
%e A131114   6, 24, 36,  24,  1;
%e A131114   6, 30, 60,  60, 30,  1;
%e A131114   6, 36, 90, 120, 90, 36, 1;
%e A131114   ...
%p A131114 seq(seq(`if`(k=n, 1, 6*binomial(n,k)), k=0..n), n=0..10); # _G. C. Greubel_, Nov 18 2019
%t A131114 Table[If[k==n, 1, 6*Binomial[n, k]], {n,0,10}, {k,0,n}]//Flatten (* _G. C. Greubel_, Nov 18 2019 *)
%o A131114 (PARI) T(n,k) = if(k==n, 1, 6*binomial(n,k)); \\ _G. C. Greubel_, Nov 18 2019
%o A131114 (Magma) [k eq n select 1 else 6*Binomial(n,k): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Nov 18 2019
%o A131114 (Sage)
%o A131114 def T(n, k):
%o A131114     if (k==n): return 1
%o A131114     else: return 6*binomial(n, k)
%o A131114 [[T(n, k) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Nov 18 2019
%o A131114 (GAP)
%o A131114 T:= function(n,k)
%o A131114     if k=n then return 1;
%o A131114     else return 6*Binomial(n,k);
%o A131114     fi;  end;
%o A131114 Flat(List([0..10], n-> List([0..n], k-> T(n,k) ))); # _G. C. Greubel_, Nov 18 2019
%Y A131114 Cf. A007318, A048488, A131110, A131111, A131112, A131113, A131115.
%K A131114 nonn,tabl,easy,less
%O A131114 0,2
%A A131114 _Gary W. Adamson_, Jun 15 2007