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A131239 Triangle, T(n,k) = 3*A007318(n,k) - 2*A046854(n,k), read by rows.

%I A131239 #8 Feb 26 2022 00:35:47
%S A131239 1,1,1,1,4,1,1,5,7,1,1,8,12,10,1,1,9,24,22,13,1,1,12,33,52,35,16,1,1,
%T A131239 13,51,85,95,51,19,1,1,16,64,148,180,156,70,22,1,1,17,88,212,348,336,
%U A131239 238,92,25,1,1,20,105,320,560,714,574,344,117,28,1,1,21,135,425,920,1274,1330,918,477,145,31,1
%N A131239 Triangle, T(n,k) = 3*A007318(n,k) - 2*A046854(n,k), read by rows.
%C A131239 Row sums = A074878: (1, 2, 6, 14, 32, 70, 239, ...).
%C A131239 A131238 = 2*A007318 - A046854.
%H A131239 G. C. Greubel, <a href="/A131239/b131239.txt">Rows n = 0..100 of triangle, flattened</a>
%F A131239 T(n,k) = 3*A007318(n,k) - 2*A046854(n,k) as infinite lower triangular matrices.
%F A131239 T(n,k) = 3*binomial(n,k) - 2*binomial(floor((n+k)/2), k). - _G. C. Greubel_, Jul 12 2019
%e A131239 First few rows of the triangle:
%e A131239   1;
%e A131239   1,  1;
%e A131239   1,  4,  1;
%e A131239   1,  5,  7,  1;
%e A131239   1,  8, 12, 10,  1;
%e A131239   1,  9, 24, 22, 13,  1;
%e A131239   1, 12, 33, 52, 35, 16, 1;
%e A131239   ...
%t A131239 With[{B=Binomial}, Table[3*B[n,k] - 2*B[Floor[(n+k)/2], k], {n,0,12}, {k,0,n}]]//Flatten (* _G. C. Greubel_, Jul 12 2019 *)
%o A131239 (PARI) b=binomial; T(n,k) = 3*b(n,k) - 2*b((n+k)\2, k);
%o A131239 for(n=0,12, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Jul 12 2019
%o A131239 (Magma) B:=Binomial; [3*B(n,k) - 2*B(Floor((n+k)/2), k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jul 12 2019
%o A131239 (Sage) b=binomial; [[3*b(n,k) - 2*b(floor((n+k)/2), k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Jul 12 2019
%o A131239 (GAP) B:=Binomial;; Flat(List([0..12], n-> List([0..n], k-> 3*B(n,k) - 2*B(Int((n+k)/2), k) ))); # _G. C. Greubel_, Jul 12 2019
%Y A131239 Cf. A007318, A046854, A131238, A074878.
%K A131239 nonn,tabl
%O A131239 0,5
%A A131239 _Gary W. Adamson_, Jun 21 2007
%E A131239 More terms added by _G. C. Greubel_, Jul 12 2019