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A273019 a(n) = hypergeom([-2*n-1, 1/2], [2], 4) + (2*n+1)*hypergeom([-n+1/2, -n], [2], 4).

%I A273019 #15 Mar 15 2020 17:36:37
%S A273019 1,5,39,321,2675,22483,190345,1621413,13882947,119385663,1030434069,
%T A273019 8921880135,77459553549,674100041501,5878674505303,51361306358401,
%U A273019 449476337521875,3939287035681807,34570459724919253,303749080936528883,2671775251987354377,23524418982229636185
%N A273019 a(n) = hypergeom([-2*n-1, 1/2], [2], 4) + (2*n+1)*hypergeom([-n+1/2, -n], [2], 4).
%F A273019 Conjecture: +(241*n+56) *(2*n+1) *(n+1)*a(n) +(482*n^3-25561*n^2+13831*n-56) *a(n-1) +(-48682*n^3+225897*n^2-300131*n+125310) *a(n-2) +9*(n-2) *(2651*n-3183) *(2*n-3) *a(n-3)=0. - _R. J. Mathar_, May 16 2016
%F A273019 Recurrence (of order 2): (n+1)*(2*n + 1)*(12*n^2 - 19*n + 9)*a(n) = (240*n^4 - 140*n^3 - 42*n^2 - 7*n + 9)*a(n-1) - 9*(n-1)*(2*n - 1)*(12*n^2 + 5*n + 2)*a(n-2). - _Vaclav Kotesovec_, Jul 05 2018
%F A273019 a(n) ~ 3^(2*n + 3/2) / (2^(3/2) * sqrt(Pi*n)). - _Vaclav Kotesovec_, Jul 05 2018
%p A273019 a := n -> hypergeom([-2*n-1, 1/2], [2], 4) + (2*n+1)*hypergeom([-n+1/2, -n], [2], 4): seq(simplify(a(n)), n=0..22);
%t A273019 Table[HypergeometricPFQ[{-2*n-1, 1/2}, {2}, 4] + (2*n+1)*HypergeometricPFQ[ {-n+1/2, -n}, {2}, 4], {n, 0, 20}] (* _Vaclav Kotesovec_, Jul 05 2018 *)
%o A273019 def A():
%o A273019     a, b, c, d, n = 0, 1, 1, -1, 1
%o A273019     while True:
%o A273019         if n%2: yield d + b*(1-(-1)^n)
%o A273019         n += 1
%o A273019         a, b = b, (3*(n-1)*n*a+(2*n-1)*n*b)//((n+1)*(n-1))
%o A273019         c, d = d, (3*(n-1)*c-(2*n-1)*d)//n
%o A273019 A273019 = A()
%o A273019 print([A273019.next() for _ in range(22)])
%Y A273019 Bisection of A273020.
%Y A273019 Cf. A082758.
%K A273019 nonn
%O A273019 0,2
%A A273019 _Peter Luschny_, May 13 2016