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%I #19 Mar 18 2019 15:39:51
%S 0,1,6,49,468,5469,73362,1138005,19737000,383284665,8163588510,
%T 190709475705,4818820261500,131650382056725,3850053335966250,
%U 120466494638624925,4002649276431128400,141156781966460192625,5252646220794868029750,206149276075766825426625
%N Expansion of the exponential generating function arcsin(2*x)/(2*(1-2*x)^(3/2)).
%C Used in A024199(n+1) = A003148(n) + a(n).
%C Binomial convolution of [0,1^2,0,2^2,0,...,0,((2*k)!/k!)^2,0,...] (e.g.f. arcsin(2*x)/2) with the double factorials A001147.
%H Robert Israel, <a href="/A143165/b143165.txt">Table of n, a(n) for n = 0..402</a>
%F E.g.f.: arcsin(2*x)/(2*(1-2*x)^(3/2)).
%F a(n) = sum(binomial(n,2*k+1)*(4^k)*((2*k-1)!!)^2*(2*(n-2*k)-1)!!,k=0..floor(n/2)), with (2*n-1)!!:= A001147(n) (double factorials).
%F a(n) ~ Pi * 2^(n-1/2) * n^(n+1) / exp(n) * (1 - sqrt(2/(Pi*n))). - _Vaclav Kotesovec_, Mar 18 2014
%F 2*(n+1)*(3+2*n)^2*a(n)-(4*n^2+8*n+1)*a(n+1)-(2*(n+4))*a(n+2)+a(n+3)=0. - _Robert Israel_, Feb 07 2018
%e a(3) + A003148(3) = 49 + 27 = 76 = A024199(4).
%p f:= gfun:-rectoproc({2*(n+1)*(3+2*n)^2*a(n)-(4*n^2+8*n+1)*a(n+1)-(2*(n+4))*a(n+2)+a(n+3)=0, a(0)=0,a(1)=1,a(2)=6},a(n),remember):
%p map(f, [$0 .. 30]); # _Robert Israel_, Feb 07 2018
%t With[{nn=20},CoefficientList[Series[ArcSin[2x]/(2(1-2x)^(3/2)),{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Mar 18 2019 *)
%o (PARI) x = 'x + O('x^40); concat(0, Vec(serlaplace(asin(2*x)/(2*(1-2*x)^(3/2))))) \\ _Michel Marcus_, Jun 18 2017
%Y Cf. A003148, A024199.
%K nonn,easy
%O 0,3
%A _Wolfdieter Lang_, Sep 15 2008
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