A159605 - OEIS

%I #8 Nov 19 2023 06:54:54

%S 1,3,63,3465,363825,62214075,15740160975,5524796502225,

%T 2569030373534625,1528573072253101875,1132672646539548489375,

%U 1022803399825212285905625,1105650475211054481063980625,1409704355894094463356575296875

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

%H Harvey P. Dale, <a href="/A159605/b159605.txt">Table of n, a(n) for n = 1..213</a>

%F a(n) = Product_{k=1..n} (2k-3)(4k-5).

%F 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 E.g.f.: A(x) = x + 3*x^3/3! + 63*x^5/5! + 3465*x^7/7! +...

%e A(S(x)) = x where S(x) = Sum_{n>=1} A159601(n)*x^(2n-1)/(2n-1)! :

%e S(x) = x - 3*x^3/3! + 27*x^5/5! - 441*x^7/7! + 11529*x^9/9! +...

%t Table[Product[(2k-3)(4k-5),{k,n}],{n,15}] (* _Harvey P. Dale_, Jan 31 2023 *)

%o (PARI) a(n)=prod(k=1,n,(2*k-3)*(4*k-5))

%Y Cf. A159601.

%K nonn

%O 1,2

%A _Paul D. Hanna_, May 11 2009