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A227624 a(n) = numerator of r(n), where r(n) = (4*n+2)!/((3*n)!*2^n).

%I A227624 #34 Jul 06 2017 00:08:28
%S A227624 2,60,1260,30030,835380,26860680,984233250,40560770250,1858741384500,
%T A227624 93814013878200,5172710627284200,309424040950649625,
%U A227624 19960884210345828750,1381474908065669917500,102111212412024699633750,8028503070893011778321250
%N A227624 a(n) = numerator of r(n), where r(n) = (4*n+2)!/((3*n)!*2^n).
%C A227624 The first values with denominators > 1 occur at n = (43, 86, 87, 91, 107, 171, 172, ...). - _G. C. Greubel_, Jul 04 2017
%H A227624 G. C. Greubel, <a href="/A227624/b227624.txt">Table of n, a(n) for n = 0..355</a>
%F A227624 In Maple notation,
%F A227624 E.g.f. of r(n):  2*hypergeom([3/4, 5/4, 3/2], [1/3, 2/3], (128/27)*z).
%F A227624 Integral representation as n-th moment of a signed function w(x) of bounded variation on (0,infinity),
%F A227624 w(x) = (5/64)*2^(3/4)*hypergeom([13/12, 17/12], [1/4, 1/2], -(27/128)*x)/(GAMMA(3/4)*x^(1/4))-(231/512)*2^(3/4)*GAMMA(3/4)*x^(1/4)*hypergeom([19/12, 23/12], [3/4, 3/2], -(27/128)*x)/Pi-(35/64)*sqrt(2)*sqrt(x)*hypergeom([11/6, 13/6], [5/4, 7/4], -(27/128)*x)/sqrt(Pi);
%F A227624 For x>5.723, w(x)>0.
%F A227624   w(0)=w(5.723)=limit(w(x),x=infinity)=0.
%F A227624   For x<5.723, w(x)<0.
%F A227624 r(n) = int(x^n*w(x), x=0..infinity), n>=0.
%F A227624 Asymptotics: r(n)-> (1/3888)*sqrt(3)*(41472*n^2+30816*n+4969)*(128/27)^n*exp(-n)*(n)^(n), for n->infinity.
%F A227624 3*(3*n-1)*(3*n-2)*r(n) -4*(4*n+1)*(2*n+1)*(4*n-1)*r(n-1)=0. - _R. J. Mathar_, Oct 08 2016
%p A227624 seq(numer(2*(4*n+2)!/((3*n)!*2^(n+1)), n=0..15)
%t A227624 Table[(4 n + 2)!/((3 n)!*2^n), {n, 0, 15}] (* _Michael De Vlieger_, Oct 08 2016 *)(* generates values of r(n) *)
%t A227624 Table[Numerator[(4*n + 2)!/((3*n)! *2^n)], {n, 0, 50}] (* _G. C. Greubel_, Jul 04 2017 *)
%o A227624 (PARI) for(n=0,50, print1(numerator((4*n + 2)!/((3*n)! *2^n)), ", ")) \\ _G. C. Greubel_, Jul 05 2017
%Y A227624 Cf. A001879, A227528.
%K A227624 nonn,easy
%O A227624 0,1
%A A227624 _Karol A. Penson_, Jul 18 2013