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%I #20 Sep 08 2022 08:45:30
%S 1,1,1,2,3,1,2,6,5,1,2,8,12,7,1,2,10,20,20,9,1,2,12,30,40,30,11,1,2,
%T 14,42,70,70,42,13,1,2,16,56,112,140,112,56,15,1,2,18,72,168,252,252,
%U 168,72,17,1,2,20,90,240,420,504,420,240,90,19,1
%N T(n,k) = 2*A007318(n,k) - A097806(n,k).
%C Row sums give A095121.
%C Triangle T(n,k), 0 <= k <= n, read by rows given by [1, 1, -2, 1, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1,0,0,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - _Philippe Deléham_, Dec 18 2007
%H G. C. Greubel, <a href="/A131108/b131108.txt">Rows n = 0..100 of triangle, flattened</a>
%F Twice Pascal's triangle minus A097806, the pairwise operator.
%F G.f.: (1-x*y+x^2+x^2*y)/((-1+x+x*y)*(x*y-1)). - _R. J. Mathar_, Aug 11 2015
%e First few rows of the triangle are:
%e 1;
%e 1, 1;
%e 2, 3, 1;
%e 2, 6, 5, 1;
%e 2, 8, 12, 7, 1;
%e 2, 10, 20, 20, 9, 1;
%e ...
%p seq(seq( `if`(k=n-1, 2*n-1, `if`(k=n, 1, 2*binomial(n,k))), k=0..n), n=0..12); # _G. C. Greubel_, Nov 18 2019
%t Table[If[k==n-1, 2*n-1, If[k==n, 1, 2*Binomial[n, k]]], {n,0,12}, {k,0, n}]//Flatten (* _G. C. Greubel_, Nov 18 2019 *)
%o (PARI) T(n,k) = if(k==n-1, 2*n-1, if(k==n, 1, 2*binomial(n,k))); \\ _G. C. Greubel_, Nov 18 2019
%o (Magma)
%o function T(n,k)
%o if k eq n-1 then return 2*n-1;
%o elif k eq n then return 1;
%o else return 2*Binomial(n,k);
%o end if;
%o return T;
%o end function;
%o [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Nov 18 2019
%o (Sage)
%o @CachedFunction
%o def T(n, k):
%o if (k==n-1): return 2*n-1
%o elif (k==n): return 1
%o else: return 2*binomial(n,k)
%o [[T(n, k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Nov 18 2019
%Y Cf. A007318, A095121, A097806.
%K nonn,tabl
%O 0,4
%A _Gary W. Adamson_, Jun 15 2007
%E Corrected by _Philippe Deléham_, Dec 17 2007
%E More terms added and data corrected by _G. C. Greubel_, Nov 18 2019
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