A362091 Expansion of odd function A(x) = Sum_{n>=1} a(n)*x^(2*n-1)/(2*n-1)! = x*Product_{odd primes p} (1 + (-x^2)^((p - 1)/2) / p).

1, -2, 24, -1056, 17280, -4769280, 727004160, -57306009600, 33169902796800, -11274754292121600, 1817252280926208000, -1974735167227232256000, 486820041614032896000000, -213735278298669436108800000, 514712400692128229464473600000, -451255060337978567952433152000000, 237712213428431126597575114752000000

Offset

1,2

Comments

Special value at x = 1: A(1) = 2/Pi = Product_{odd primes p} (1 + (-1)^((p - 1)/2) / p), due to Euler.

Links

Table of n, a(n) for n=1..17.

Example

E.g.f. A(x) = x - 2*x^3/3! + 24*x^5/5! - 1056*x^7/7! + 17280*x^9/9! - 4769280*x^11/11! + 727004160*x^13/13! - 57306009600*x^15/15! + 33169902796800*x^17/17! - 11274754292121600*x^19/19! + ...
where
A(x) = x*(1 - x^2/3)*(1 + x^4/5)*(1 - x^6/7)*(1 - x^10/11)*(1 + x^12/13)*(1 + x^16/17)*(1 - x^18/19)*(1 - x^22/23)*(1 + x^28/29)*(1 - x^30/31) * ... * (1 + (-x^2)^((prime(n) - 1)/2)/prime(n)) * ...
As a power series in x with reduced fractional coefficients,
A(x) = x - 1/3*x^3 + 1/5*x^5 - 22/105*x^7 + 1/21*x^9 - 46/385*x^11 + 1753/15015*x^13 - 94/2145*x^15 + 2164/23205*x^17 - 149837/1616615*x^19 + 172504/4849845*x^21 - 8520599/111546435*x^23 + 700178/22309287*x^25 - 2189521/111546435*x^27 + 188312654/3234846615*x^29 - 1834401717/33426748355*x^31 + ...

Prog

(PARI) {a(n) = my(A = x*prod(m=2, n, 1 + (-x^2)^((prime(m) - 1)/2)/prime(m) +O(x^(2*n+1)))); (2*n-1)! * polcoeff(A, 2*n-1)}
for(n=1, 20, print1(a(n), ", "))

Crossrefs

Sequence in context: A213984 A129622 A268311 * A330087 A357827 A307157

Adjacent sequences: A362088 A362089 A362090 * A362092 A362093 A362094

Keyword

sign

Author

Paul D. Hanna, Apr 08 2023

Status

approved