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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)).

%I A380570 #40 Feb 05 2025 22:17:38
%S A380570 1,4,-1,16,-40,9,64,-560,1036,-225,256,-5376,31584,-51664,11025,1024,
%T A380570 -42240,561792,-2764960,4228884,-893025,4096,-292864,7358208,
%U A380570 -79036672,351475696,-515267064,108056025,16384,-1863680,78926848,-1559683840,14763100352,-61460460880,87512357916
%N 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)).
%C A380570 Odd coefficients of x are excluded here because they are zero.
%H A380570 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HankelsSymbol.html">Hankel's Symbol</a>.
%F A380570 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)..
%F A380570 T(n, k) = A008956(n, k)*4^(n-k)*(-1)^k.
%F A380570 Sum_{k=0..n} T(n, k) = A380612(n) = -(-4)^n*Gamma(-1/2 + n)*Gamma(3/2 + n)/Pi.
%e A380570 Triangle begins:
%e A380570  n \ k: 0        1        2          3          4           5          6
%e A380570       x^0      x^2      x^4        x^6        x^8        x^10       x^12
%e A380570 [0]     1;
%e A380570 [1]     4,      -1;
%e A380570 [2]    16,     -40,       9;
%e A380570 [3]    64,    -560,    1036,      -225;
%e A380570 [4]   256,   -5376,   31584,    -51664,     11025;
%e A380570 [5]  1024,  -42240,  561792,  -2764960,   4228884,    -893025;
%e A380570 [6]  4096, -292864, 7358208, -79036672, 351475696, -515267064, 108056025;
%e A380570      ...
%o A380570 (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]
%Y A380570 Cf. A000302 (column 0).
%Y A380570 Cf. A001818 (absolute values of main diagonal).
%Y A380570 Cf. A001824 (1/4 of absolute values of second diagonal).
%Y A380570 Cf. A001825 (1/16 of absolute values of second diagonal).
%Y A380570 Cf. A380612 (row sums).
%Y A380570 Cf. A008956.
%K A380570 sign,tabl
%O A380570 0,2
%A A380570 _Thomas Scheuerle_, Jan 27 2025