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A159605 E.g.f: Sum_{n>=1} a(n)*x^(2n-1)/(2n-1)! = Series_Reversion of e.g.f. S(x) of A159601.

%I A159605 #8 Nov 19 2023 06:54:54
%S A159605 1,3,63,3465,363825,62214075,15740160975,5524796502225,
%T A159605 2569030373534625,1528573072253101875,1132672646539548489375,
%U A159605 1022803399825212285905625,1105650475211054481063980625,1409704355894094463356575296875
%N A159605 E.g.f: Sum_{n>=1} a(n)*x^(2n-1)/(2n-1)! = Series_Reversion of e.g.f. S(x) of A159601.
%H A159605 Harvey P. Dale, <a href="/A159605/b159605.txt">Table of n, a(n) for n = 1..213</a>
%F A159605 a(n) = Product_{k=1..n} (2k-3)(4k-5).
%F A159605 a(n) ~ Gamma(1/4) * 2^(3*n - 5/2) * n^(2*n - 7/4) / (sqrt(Pi) * exp(2*n)). - _Vaclav Kotesovec_, Nov 19 2023
%e A159605 E.g.f.: A(x) = x + 3*x^3/3! + 63*x^5/5! + 3465*x^7/7! +...
%e A159605 A(S(x)) = x where S(x) = Sum_{n>=1} A159601(n)*x^(2n-1)/(2n-1)! :
%e A159605 S(x) = x - 3*x^3/3! + 27*x^5/5! - 441*x^7/7! + 11529*x^9/9! +...
%t A159605 Table[Product[(2k-3)(4k-5),{k,n}],{n,15}] (* _Harvey P. Dale_, Jan 31 2023 *)
%o A159605 (PARI) a(n)=prod(k=1,n,(2*k-3)*(4*k-5))
%Y A159605 Cf. A159601.
%K A159605 nonn
%O A159605 1,2
%A A159605 _Paul D. Hanna_, May 11 2009