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A141597 Triangle T(n,k) = 2*binomial(n,k)^2 - 1, read by rows, 0<=k<=n.

%I A141597 #12 Sep 15 2024 06:49:48
%S A141597 1,1,1,1,7,1,1,17,17,1,1,31,71,31,1,1,49,199,199,49,1,1,71,449,799,
%T A141597 449,71,1,1,97,881,2449,2449,881,97,1,1,127,1567,6271,9799,6271,1567,
%U A141597 127,1,1,161,2591,14111,31751,31751,14111,2591,161,1,1,199,4049,28799,88199,127007,88199,28799,4049,199,1
%N A141597 Triangle T(n,k) = 2*binomial(n,k)^2 - 1, read by rows, 0<=k<=n.
%H A141597 G. C. Greubel, <a href="/A141597/b141597.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A141597 Sum_{k=0..n} T(n, k) = A134759(n) = 2*binomial(2*n,n) - (n+1) (row sums).
%F A141597 T(n, n-k) = T(n, k).
%F A141597 Sum_{k=0..n} (-1)^k*T(n, k) = ((1+(-1)^n)/2)*(2*(-1)^(n/2)*binomial(n, n/2) - 1) (alternating sign row sums). - _G. C. Greubel_, Sep 15 2024
%e A141597 Triangle begins as:
%e A141597   1;
%e A141597   1,   1;
%e A141597   1,   7,    1;
%e A141597   1,  17,   17,     1;
%e A141597   1,  31,   71,    31,     1;
%e A141597   1,  49,  199,   199,    49,      1;
%e A141597   1,  71,  449,   799,   449,     71,     1;
%e A141597   1,  97,  881,  2449,  2449,    881,    97,     1;
%e A141597   1, 127, 1567,  6271,  9799,   6271,  1567,   127,    1;
%e A141597   1, 161, 2591, 14111, 31751,  31751, 14111,  2591,  161,   1;
%e A141597   1, 199, 4049, 28799, 88199, 127007, 88199, 28799, 4049, 199,  1;
%t A141597 T[n_, m_, k_, l_]:= (1+l)*Binomial[n, m]^k -l;
%t A141597 Table[T[n,m,2,1], {n,0,12}, {m,0,n}]//Flatten
%o A141597 (Magma)
%o A141597 A141597:= func< n,k | 2*Binomial(n,k)^2 - 1 >;
%o A141597 [A141597(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Sep 15 2024
%o A141597 (SageMath)
%o A141597 def A141597(n,k): return 2*binomial(n,k)^2 -1
%o A141597 flatten([[A141597(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Sep 15 2024
%Y A141597 Cf. A134759 (row sums), A141596.
%K A141597 nonn,tabl
%O A141597 0,5
%A A141597 _Roger L. Bagula_ and _Gary W. Adamson_, Aug 21 2008