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%I #12 May 31 2013 10:11:59
%S 1,2,6,36,252,2520,27720,388080,5821200,104781600,1990850400,
%T 43798708800,1007370302400,26191627862400,707173952284800,
%U 21215218568544000,657671775624864000,22360840371245376000,782629412993588160000,29739917693756350080000
%N a(n) = 2*n*a(n-1) if the parity of the ratio a(n-1)/a(n-2) is odd, otherwise (for even parity) a(n) = (2n-1)*a(n-1).
%C The sequence is designed to be factorial like in even/ odd 2*n and 2*n-1.
%H Vincenzo Librandi, <a href="/A177373/b177373.txt">Table of n, a(n) for n = 0..200</a>
%F a(n) = 2*(a(n-1)+(n-1)*(2n-3)*a(n-2)). - _R. J. Mathar_, Jan 28 2012
%F E.g.f.: (1+2*x)^(1/4)/(1-2*x)^(3/4). - _Vaclav Kotesovec_, Oct 21 2012
%F a(n) ~ n!*2^(n+1/4)/(Gamma(3/4)*n^(1/4)). - _Vaclav Kotesovec_, Oct 21 2012
%e The parity of 2/1 = 2 is even, so a(2) = (2*2-1)*2 = 6. The parity of 6/2=3 is odd, so a(3) = (2*3)*6 = 36.
%t w[ -1] := 1; w[0] := 1;
%t w[n_] := w[n] = If[Mod[ w[n - 1]/w[n - 2], 2] == 0,
%t (2* n - 1)*w[n - 1], If[Mod[w[n - 1]/w[n - 2], 2] == 1, 2*n*w[n - 1], 0]] Table[w[n], {n, 0, 20}]
%t CoefficientList[Series[(1+2*x)^(1/4)/(1-2*x)^(3/4),{x,0,20}],x] *Range[0, 20]! (* _Vaclav Kotesovec_, Oct 21 2012 *)
%Y Cf. A042948.
%K nonn,easy
%O 0,2
%A _Roger L. Bagula_, May 07 2010
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