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A381059 Array read by ascending antidiagonals: A(n,k) = numerator(binomial(n-1/2,k)) with k >=0.

%I A381059 #14 Feb 17 2025 03:20:47
%S A381059 1,1,-1,1,1,3,1,3,-1,-5,1,5,3,1,35,1,7,15,-1,-5,-63,1,9,35,5,3,7,231,
%T A381059 1,11,63,35,-5,-3,-21,-429,1,13,99,105,35,3,7,33,6435,1,15,143,231,
%U A381059 315,-7,-5,-9,-429,-12155,1,17,195,429,1155,63,7,5,99,715,46189
%N A381059 Array read by ascending antidiagonals: A(n,k) = numerator(binomial(n-1/2,k)) with k >=0.
%C A381059 Numerators of the binomial coefficients for half-integers. The denominators are given by the absolute values of A173755.
%H A381059 Stefano Spezia, <a href="/A381059/b381059.txt">Table of n, a(n) for n = 0..11324</a> (first 150 antidiagonals of the array)
%H A381059 Gábor Hegedüs, Sho Suda, and Ziqing Xiang, <a href="https://arxiv.org/abs/2501.11461">Codes with symmetric distances</a>, arXiv:2501.11461 [math.CO], 2025. See p. 10.
%F A381059 A(n,k) = numerator((2*n - 1)!!/((2*(n - k) - 1)!!*2^k*k!)).
%F A381059 A(n,2) = A000466(n-1) for n > 0.
%F A381059 A(n,3) = A162540(n-3) for n > 3.
%F A381059 A(0,n) = (-1)^n*A001790(n).
%F A381059 abs(A(2,n)) = abs(A161200(n)).
%F A381059 abs(A(3,n)) = abs(A161202(n)).
%e A381059 The array of the binomial coefficients for half-integers begins as:
%e A381059   1, -1/2,  3/8,  -5/16,   35/128, -63/256, ...
%e A381059   1,  1/2, -1/8,   1/16,   -5/128,   7/256, ...
%e A381059   1,  3/2,  3/8,  -1/16,    3/128,  -3/256, ...
%e A381059   1,  5/2, 15/8,   5/16,   -5/128,   3/256, ...
%e A381059   1,  7/2, 35/8,  35/16,   35/128,  -7/256, ...
%e A381059   1,  9/2, 63/8, 105/16,  315/128,  63/256, ...
%e A381059   1, 11/2, 99/8, 231/16, 1155/128, 693/256, ...
%e A381059   ...
%t A381059 A[n_,k_]:=Numerator[Binomial[n-1/2,k]]; Table[A[n-k,k],{n,0,10},{k,0,n}]//Flatten (* or *)
%t A381059 A[n_,k_]:=Numerator[(2n-1)!!/((2(n-k)-1)!!2^k k!)]; Table[A[n-k,k],{n,0,10},{k,0,n}]//Flatten
%Y A381059 Cf. A000466, A161200, A161202, A162540, A173755.
%Y A381059 Columns k=0..1 give A000012, A060747.
%Y A381059 Row n=1 gives A002596.
%Y A381059 Main diagonal gives A001790.
%K A381059 sign,frac,look,tabl
%O A381059 0,6
%A A381059 _Stefano Spezia_, Feb 12 2025