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A141901 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.

%I A141901 #10 Jan 31 2025 22:22:40
%S A141901 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 A141901 -1,1,56,67,48,21,4,-1,-1,1,119,155,131,77,29,5,-1,-1,1,246,338,318,
%U A141901 224,114,38,6,-1,-1,1,501,712,720,574,354,160,48,7,-1,-1
%N A141901 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 A141901 G. C. Greubel, <a href="/A141901/b141901.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A141901 T(n, k) = binomial(n, k+1)*Hypergeometric2F1([1, 1+k-n], [2+k], -1) - 2^(n-k) with T(n, 0) = 1.
%F A141901 From _G. C. Greubel_, Mar 29 2021: (Start)
%F A141901 T(n, k) = Sum_{j=0..n-k-1} binomial(n, j+k+1) - 2^(n-k) with T(n, 0) = 1.
%F A141901 Sum_{k=0..n} T(n, k) = 2^(n-1)*(n-4) + 3 = A036799(n-3) - A000225(n-1). (End)
%e A141901 Triangle begins as:
%e A141901   1;
%e A141901   1,  -1;
%e A141901   1,  -1,  -1;
%e A141901   1,   0,  -1,  -1;
%e A141901   1,   3,   1,  -1,  -1;
%e A141901   1,  10,   8,   2,  -1,  -1;
%e A141901   1,  25,  26,  14,   3,  -1,  -1;
%e A141901   1,  56,  67,  48,  21,   4,  -1, -1;
%e A141901   1, 119, 155, 131,  77,  29,   5, -1, -1;
%e A141901   1, 246, 338, 318, 224, 114,  38,  6, -1, -1;
%e A141901   1, 501, 712, 720, 574, 354, 160, 48,  7, -1, -1;
%t A141901 (* First program *)
%t A141901 T[n_, k_]:= If[k==0, 1, Binomial[n,k+1]*Hypergeometric2F1[1, 1+k-n, 2+k, -1] - 2^(n-k)];
%t A141901 Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Mar 29 2021 *)
%t A141901 (* Second program *)
%t A141901 T[n_, k_]:= If[k==0, 1, Sum[Binomial[n, j+k+1], {j, 0, n-k-1}] - 2^(n-k)];
%t A141901 Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* _G. C. Greubel_, Mar 29 2021 *)
%o A141901 (Magma)
%o A141901 T:= func< n,k | k eq 0 select 1 else (&+[Binomial(n, j+k+1): j in [0..n]]) - 2^(n-k)>;
%o A141901 [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Mar 29 2021
%o A141901 (SageMath)
%o A141901 @CachedFunction
%o A141901 def T(n,k):
%o A141901     if k==0: return 1
%o A141901     else: return sum( binomial(n, j+k+1) for j in (0..n-k-1) ) - 2^(n-k)
%o A141901 flatten([[T(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Mar 29 2021
%Y A141901 Cf. A000225, A036799.
%K A141901 sign,tabl
%O A141901 0,12
%A A141901 _Roger L. Bagula_ and _Gary W. Adamson_, Sep 13 2008
%E A141901 Edited by _G. C. Greubel_, Mar 29 2021