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A131108 T(n,k) = 2*A007318(n,k) - A097806(n,k).

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