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A131238 Triangle read by rows: T(n,k) = 2*binomial(n,k) - binomial(floor((n+k)/2), k) (0 <= k <= n).

%I A131238 #9 Feb 26 2022 00:35:22
%S A131238 1,1,1,1,3,1,1,4,5,1,1,6,9,7,1,1,7,17,16,9,1,1,9,24,36,25,11,1,1,10,
%T A131238 36,60,65,36,13,1,1,12,46,102,125,106,49,15,1,1,13,62,148,237,231,161,
%U A131238 64,17,1,1,15,75,220,385,483,392,232,81,19,1,1,16,95,295,625,868,896,624,321,100,21,1
%N A131238 Triangle read by rows: T(n,k) = 2*binomial(n,k) - binomial(floor((n+k)/2), k) (0 <= k <= n).
%C A131238 Row sums = A027934: (1, 2, 5, 11, 24, 51, 107, ...).
%C A131238 A131239 = 3*A007318 - 2*A046854.
%H A131238 G. C. Greubel, <a href="/A131238/b131238.txt">Rows n = 0..100 of triangle, flattened</a>
%F A131238 T(n,k) = 2*A007318(n,k) - A046854(n,k) as infinite lower triangular matrices, where A007318 = Pascal's triangle and A046854 = Pascal's triangle with repeats, by columns.
%e A131238 First few rows of the triangle:
%e A131238   1;
%e A131238   1,  1;
%e A131238   1,  3,  1;
%e A131238   1,  4,  5,  1;
%e A131238   1,  6,  9,  7,  1;
%e A131238   1,  7, 17, 16,  9,  1;
%e A131238   1,  9, 24, 36, 25, 11,  1;
%e A131238   1, 10, 36, 60, 65, 36, 13, 1;
%e A131238   ...
%p A131238 T := proc (n, k) options operator, arrow; 2*binomial(n, k)-binomial(floor((1/2)*n+(1/2)*k), k) end proc: for n from 0 to 9 do seq(T(n, k), k = 0 .. n) end do; # yields sequence in triangular form - _Emeric Deutsch_, Jul 09 2007
%t A131238 With[{B = Binomial}, Table[2*B[n, k] - B[Floor[(n+k)/2], k], {n,0,12}, {k,0,n}]]//Flatten (* _G. C. Greubel_, Jul 12 2019 *)
%o A131238 (PARI) b=binomial; T(n,k) = 2*b(n,k) - b((n+k)\2, k);
%o A131238 for(n=0,12, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Jul 12 2019
%o A131238 (Magma) B:=Binomial; [2*B(n,k) - B(Floor((n+k)/2), k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jul 12 2019
%o A131238 (Sage) b=binomial; [[2*b(n,k) - b(floor((n+k)/2), k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Jul 12 2019
%o A131238 (GAP) B:=Binomial;; Flat(List([0..12], n-> List([0..n], k-> 2*B(n,k) - B(Int((n+k)/2), k) ))); # _G. C. Greubel_, Jul 12 2019
%Y A131238 Cf. A027934, A131239, A007318, A046854.
%K A131238 nonn,tabl
%O A131238 0,5
%A A131238 _Gary W. Adamson_, Jun 21 2007
%E A131238 More terms added by _G. C. Greubel_, Jul 12 2019