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%I #7 Mar 30 2021 01:48:28
%S 1,1,-1,1,-1,-1,1,0,-1,-1,1,3,1,-1,-1,1,10,8,2,-1,-1,1,25,26,14,3,-1,
%T -1,1,56,67,48,21,4,-1,-1,1,119,155,131,77,29,5,-1,-1,1,246,338,318,
%U 224,114,38,6,-1,-1,1,501,712,720,574,354,160,48,7,-1,-1
%N Triangle T(n, k) = Sum_{j=0..n-k-1} binomial(n, j+k+1) - 2^(n-k) with T(n, 0) = 1, read by rows.
%H G. C. Greubel, <a href="/A141901/b141901.txt">Rows n = 0..50 of the triangle, flattened</a>
%F T(n, k) = binomial(n, k+1)*Hypergeometric2F1([1, 1+k-n], [2+k], -1) - 2^(n-k) with T(n, 0) = 1.
%F From _G. C. Greubel_, Mar 29 2021: (Start)
%F T(n, k) = Sum_{j=0..n-k-1} binomial(n, j+k+1) - 2^(n-k) with T(n, 0) = 1.
%F Sum_{k=0..n} T(n, k) = 2^(n-1)*(n-4) + 3 = A036799(n-3) - A000225(n-1). (End)
%e Triangle begins as:
%e 1;
%e 1, -1;
%e 1, -1, -1;
%e 1, 0, -1, -1;
%e 1, 3, 1, -1, -1;
%e 1, 10, 8, 2, -1, -1;
%e 1, 25, 26, 14, 3, -1, -1;
%e 1, 56, 67, 48, 21, 4, -1, -1;
%e 1, 119, 155, 131, 77, 29, 5, -1, -1;
%e 1, 246, 338, 318, 224, 114, 38, 6, -1, -1;
%e 1, 501, 712, 720, 574, 354, 160, 48, 7, -1, -1;
%t (* First program *)
%t T[n_, k_]:= If[k==0, 1, Binomial[n,k+1]*Hypergeometric2F1[1, 1+k-n, 2+k, -1] - 2^(n-k)];
%t Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Mar 29 2021 *)
%t (* Second program *)
%t T[n_, k_]:= If[k==0, 1, Sum[Binomial[n, j+k+1], {j, 0, n-k-1}] - 2^(n-k)];
%t Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* _G. C. Greubel_, Mar 29 2021 *)
%o (Magma)
%o T:= func< n,k | k eq 0 select 1 else (&+[Binomial(n, j+k+1): j in [0..n]]) - 2^(n-k)>;
%o [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Mar 29 2021
%o (Sage)
%o @CachedFunction
%o def T(n,k):
%o if (k==0: return 1
%o else: return sum( binomial(n, j+k+1) for j in (0..n-k-1) ) - 2^(n-k)
%o flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Mar 29 2021
%Y Cf. A000225, A036799.
%K sign,tabl
%O 0,12
%A _Roger L. Bagula_ and _Gary W. Adamson_, Sep 13 2008
%E Edited by _G. C. Greubel_, Mar 29 2021
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