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A131240 T(n,k) = 2*A046854(n,k) - I.

%I A131240 #11 Feb 26 2022 00:36:07
%S A131240 1,2,1,2,2,1,2,4,2,1,2,4,6,2,1,2,6,6,8,2,1,2,6,12,8,10,2,1,2,8,12,20,
%T A131240 10,12,2,1,2,8,20,20,30,12,14,2,1,2,10,20,40,30,42,14,16,2,1,2,10,30,
%U A131240 40,70,42,56,16,18,2,1,2,12,30,70,70,112,56,72,18,20,2,1
%N A131240 T(n,k) = 2*A046854(n,k) - I.
%C A131240 Row sums = A001595: (1, 3, 5, 9, 15, 25, 41, 67, ...).
%C A131240 A131241 = 3*A046854 - 2*I.
%H A131240 G. C. Greubel, <a href="/A131240/b131240.txt">Rows n = 0..100 of triangle, flattened</a>
%F A131240 T(n,k) = 2*A046854(n,k) - Identity matrix, where A046854 = Pascal's triangle with repeats by columns.
%e A131240 First few rows of the triangle:
%e A131240   1;
%e A131240   2, 1;
%e A131240   2, 2,  1;
%e A131240   2, 4,  2, 1;
%e A131240   2, 4,  6, 2,  1;
%e A131240   2, 6,  6, 8,  2, 1;
%e A131240   2, 6, 12, 8, 10, 2, 1;
%e A131240   ...
%t A131240 Table[If[k==n, 1, 2*Binomial[Floor[(n+k)/2], k]], {n, 0, 12}, {k, 0, n}]//Flatten (* _G. C. Greubel_, Jul 12 2019 *)
%o A131240 (PARI) T(n,k) = if(k==n, 1, 2*binomial((n+k)\2, k));
%o A131240 (Magma) [k eq n select 1 else 2*Binomial(Floor((n+k)/2), k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jul 12 2019
%o A131240 (Sage)
%o A131240 def T(n, k):
%o A131240     if (k==n): return 1
%o A131240     else: return 2*binomial(floor((n+k)/2), k)
%o A131240 [[T(n, k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Jul 12 2019
%o A131240 (GAP)
%o A131240 T:= function(n,k)
%o A131240     if k=n then return 1;
%o A131240     else return 2*Binomial(Int((n+k)/2), k);
%o A131240     fi;
%o A131240   end;
%o A131240 Flat(List([0..12], n-> List([0..n], k-> T(n,k)))); # _G. C. Greubel_, Jul 12 2019
%Y A131240 Cf. A001595, A046854, A131241.
%K A131240 nonn,tabl
%O A131240 0,2
%A A131240 _Gary W. Adamson_, Jun 21 2007
%E A131240 More terms added by _G. C. Greubel_, Jul 12 2019