A357228 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, 27, 1095, 100905, 11189745, 2378802195, 524908799415, 186506150655825, 72527385885379425, 38034576658499496075, 21992048437363887457575, 16748861395227762355580025, 13415760683905948372840460625, 13429242464029329763489941151875, 14657793954450002863353646629204375

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(S(x)) = x and S(A(x)) = x, where S(x) is described by A357550.

Example

E.g.f.: A(x) = x + x^3/3! + 27*x^5/5! + 1095*x^7/7! + 100905*x^9/9! + 11189745*x^11/11! + 2378802195*x^13/13! + 524908799415*x^15/15! + 186506150655825*x^17/17! + ...
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! - 17*x^5/5! + 137*x^7/7! + 13009*x^9/9! + 3098111*x^11/11! - 499973633*x^13/13! + 13063051433*x^15/15! - 12602400051359*x^17/17! + ... + A357550(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. A357229, A357550.

Sequence in context: A159457 A290946 A223552 * A104206 A327596 A163568

Adjacent sequences: A357225 A357226 A357227 * A357229 A357230 A357231

Keyword

nonn

Author

Paul D. Hanna, Oct 02 2022

Status

approved