OFFSET
1,3
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..300
FORMULA
The generating function A(x) = Sum_{n>=1} a(n) * x^(2*n-1)/(2*n-1)! satisfies the following.
(1) A(x) = Integral Product_{n>=1} 1/(1 + x^(2*n))^((2*n-1)/(2*n)) dx.
(2) A(x) = Integral Product_{n>=1} (1 + x^(2*n))^(1/(2*n)) * (1 - x^(4*n-2)) dx.
(3) A(S(x)) = x and S(A(x)) = x, where S(x) is described by A357230.
EXAMPLE
E.g.f.: A(x) = x - x^3/3! - 9*x^5/5! - 555*x^7/7! + 7665*x^9/9! - 1777545*x^11/11! + 114147495*x^13/13! - 27004972995*x^15/15! + 20805419059425*x^17/17! - 4204053743915025*x^19/19! + ...
where
d/dx A(x) = 1/( (1 + x^2)^(1/2) * (1 + x^4)^(3/4) * (1 + x^6)^(5/6) * (1 + x^8)^(7/8) * (1 + x^10)^(9/10) * ... * (1 + x^(2*n))^((2*n-1)/(2*n)) * ... ).
RELATED SERIES.
Let S(x) be the series reversion of A(x) so that A(S(x)) = x, then S(x) begins:
S(x) = x + x^3/3! + 19*x^5/5! + 1339*x^7/7! + 126121*x^9/9! + 22936441*x^11/11! + 6074972299*x^13/13! + 2211448022179*x^15/15! + ... + A357230(n)*x^(2*n-1)/(2*n-1)! + ...
PROG
(PARI) {a(n) = my(A); A = intformal( prod(k=1, n, 1/(1 + x^(2*k) + O(x^(2*n)) )^((2*k-1)/(2*k)) ) ); (2*n-1)! * polcoeff(A, 2*n-1)}
for(n=1, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Oct 02 2022
STATUS
approved