A357229 a(n) = coefficient of x^(2*n-1)/(2*n-1)! in the odd function A(x) = Integral Product_{n>=1} 1/(1 + x^(2*n))^((2*n-1)/(2*n)) dx.

1, -1, -9, -555, 7665, -1777545, 114147495, -27004972995, 20805419059425, -4204053743915025, 1822343877322543575, -505299954078654810075, 786572202448438396815825, -304143708374573670923945625, 297888516150523156788428874375, -379957456647051856809688318741875

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

Cf. A357230, A357228.

Sequence in context: A269661 A287043 A317347 * A267548 A326614 A007324

Adjacent sequences: A357226 A357227 A357228 * A357230 A357231 A357232

Keyword

sign

Author

Paul D. Hanna, Oct 02 2022

Status

approved