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%I #13 Sep 08 2022 08:45:28
%S 1,2,-1,8,-8,3,48,-72,54,-15,384,-768,864,-480,105,3840,-9600,14400,
%T -12000,5250,-945,46080,-138240,259200,-288000,189000,-68040,10395,
%U 645120,-2257920,5080320,-7056000,6174000,-3333960,1018710,-135135,10321920,-41287680,108380160,-180633600,197568000
%N Triangle read by rows: T(n,k) = (-1)^k * n! * 2^(n-2*k) * binomial(n,k) * binomial(2*k,k) (0<=k<=n).
%C Row sums yield the double factorial numbers (A001147).
%D B. T. Gill, Math. Magazine, vol. 79, No. 4, 2006, p. 313, problem 1729.
%H G. C. Greubel, <a href="/A123516/b123516.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F T(n,0) = 2^n * n! = A000165(n).
%F T(n,n) = (-1)^n*A001147(n).
%e Triangle begins:
%e 1;
%e 2, -1;
%e 8, -8, 3;
%e 48, -72, 54, -15;
%e 384, -768, 864, -480, 105;
%e 3840, -9600, 14400, -12000, 5250, -945;
%e 46080, -138240, 259200, -288000, 189000, -68040, 10395;
%e ...
%p T:=(n,k)->(-1)^k*n!*2^(n-2*k)*binomial(n,k)*binomial(2*k,k): for n from 0 to 8 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form
%t Table[(-1)^k*n! 2^(n - 2 k)*Binomial[n, k]*Binomial[2*k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _G. C. Greubel_, Oct 14 2017 *)
%o (PARI) for(n=0,10, for(k=0,n, print1((-1)^k*n!*2^(n-2*k)*binomial(n,k)* binomial(2*k,k), ", "))) \\ _G. C. Greubel_, Oct 14 2017
%o (Magma) /* As triangle * / [[(-1)^k*Factorial(n)*2^(n-2*k)* Binomial(n,k)*Binomial(2*k,k): k in [0..n]]: n in [0.. 15]]; // _Vincenzo Librandi_, Oct 15 2017
%Y Cf. A001147, A000165.
%K sign,tabl
%O 0,2
%A _Emeric Deutsch_, Oct 14 2006
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