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

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, 1, 11, 63, 35, -5, -3, -21, -429, 1, 13, 99, 105, 35, 3, 7, 33, 6435, 1, 15, 143, 231, 315, -7, -5, -9, -429, -12155, 1, 17, 195, 429, 1155, 63, 7, 5, 99, 715, 46189

Offset: 0

Author

Stefano Spezia, Feb 12 2025

Keywords

• sign
• frac
• look
• tabl

Comments

Numerators of the binomial coefficients for half-integers. The denominators are given by the absolute values of A173755.

Examples

			The array of the binomial coefficients for half-integers begins as:
  1, -1/2,  3/8,  -5/16,   35/128, -63/256, ...
  1,  1/2, -1/8,   1/16,   -5/128,   7/256, ...
  1,  3/2,  3/8,  -1/16,    3/128,  -3/256, ...
  1,  5/2, 15/8,   5/16,   -5/128,   3/256, ...
  1,  7/2, 35/8,  35/16,   35/128,  -7/256, ...
  1,  9/2, 63/8, 105/16,  315/128,  63/256, ...
  1, 11/2, 99/8, 231/16, 1155/128, 693/256, ...
  ...
		

Links

• Stefano Spezia, Table of n, a(n) for n = 0..11324 (first 150 antidiagonals of the array)
• Gábor Hegedüs, Sho Suda, and Ziqing Xiang, Codes with symmetric distances, arXiv:2501.11461 [math.CO], 2025. See p. 10.

Crossrefs

Cf. A000466, A161200, A161202, A162540, A173755.

Columns k=0..1 give A000012, A060747.

Row n=1 gives A002596.

Main diagonal gives A001790.

Programs

Mathematica

A[n_,k_]:=Numerator[Binomial[n-1/2,k]]; Table[A[n-k,k],{n,0,10},{k,0,n}]//Flatten (* or *)
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

Formula

A(n,k) = numerator((2*n - 1)!!/((2*(n - k) - 1)!!*2^k*k!)).

A(n,2) = A000466(n-1) for n > 0.

A(n,3) = A162540(n-3) for n > 3.

A(0,n) = (-1)^n*A001790(n).

abs(A(2,n)) = abs(A161200(n)).

abs(A(3,n)) = abs(A161202(n)).