A380570 Triangle T(n, k) read by rows: Row n gives the coefficients of the even powers in Product_{t=1..n}(2*x - (2*t - 1))*Product_{t=1..n}(2*x + (2*t - 1)).

1, 4, -1, 16, -40, 9, 64, -560, 1036, -225, 256, -5376, 31584, -51664, 11025, 1024, -42240, 561792, -2764960, 4228884, -893025, 4096, -292864, 7358208, -79036672, 351475696, -515267064, 108056025, 16384, -1863680, 78926848, -1559683840, 14763100352, -61460460880, 87512357916

Offset: 0

Author

Thomas Scheuerle, Jan 27 2025

Keywords

• sign
• tabl

Comments

Odd coefficients of x are excluded here because they are zero.

Examples

			Triangle begins:
 n \ k: 0        1        2          3          4           5          6
      x^0      x^2      x^4        x^6        x^8        x^10       x^12
[0]     1;
[1]     4,      -1;
[2]    16,     -40,       9;
[3]    64,    -560,    1036,      -225;
[4]   256,   -5376,   31584,    -51664,     11025;
[5]  1024,  -42240,  561792,  -2764960,   4228884,    -893025;
[6]  4096, -292864, 7358208, -79036672, 351475696, -515267064, 108056025;
     ...
		

Links

• Eric Weisstein's World of Mathematics, Hankel's Symbol.

Crossrefs

Cf. A000302 (column 0).

Cf. A001818 (absolute values of main diagonal).

Cf. A001824 (1/4 of absolute values of second diagonal).

Cf. A001825 (1/16 of absolute values of second diagonal).

Cf. A380612 (row sums).

Cf. A008956.

Programs

PARI

T(n, k) = Vec(prod(k=1,n,2*x-(2*k-1))*prod(k=1,n,2*x+(2*k-1)))[1+2*k]

Formula

The Hankel symbol (x, n) is defined as (-1)^n*cos(Pi*x)*Gamma(1/2+n-x)*Gamma(1/2+n+x)/(Pi*n!) = (cos(Pi*x)/((-4)^n*n!))*Sum_{k=0..n} T(n, k)*x^(2*k)..

T(n, k) = A008956(n, k)*4^(n-k)*(-1)^k.

Sum_{k=0..n} T(n, k) = A380612(n) = -(-4)^n*Gamma(-1/2 + n)*Gamma(3/2 + n)/Pi.