A357550 - OEIS

%I #13 Dec 03 2022 12:01:38

%S 1,-1,-17,137,13009,3098111,-499973633,13063051433,-12602400051359,

%T 1142264265564479,4900244939751731023,-1617265022962564577143,

%U -876540661492989775332431,772526162637086182379155391,-84757568544981649947240558113,-969537581289651588574578501447127

%N a(n) = coefficient of x^(2*n-1)/(2*n-1)! in the expansion of the odd function S(x) defined by: S(x) = Integral Product_{n>=1} C(n,x)^(2*n-1) dx, where C(n,x) = (1 - S(x)^(2*n))^(1/(2*n)) for n >= 1.

%H Paul D. Hanna, <a href="/A357550/b357550.txt">Table of n, a(n) for n = 1..300</a>

%F Odd function S(x) = Sum_{n>=1} a(n) * x^(2*n-1)/(2*n-1)! and related even functions C(n,x), n >= 1, satisfy the following relations.

%F (1.a) C(n,x)^(2*n) + S(x)^(2*n) = 1, for n >= 1.

%F (1.b) C'(n,x) * C(n,x)^(2*n-1) = -S'(x) * S(x)^(2*n-1), for n >= 1.

%F (2.a) S(x) = Integral Product_{n>=1} C(n,x)^(2*n-1) dx.

%F (2.b) C(n,x) = 1 - Integral S(x)^(2*n-1)/C(n,x)^(2*n-1) * Product_{k>=1} C(k,x)^(2*k-1) dx, for n >= 1.

%F (3) C(n,x)^n + i*S(x)^n = exp( n*i * Integral S'(x)*S(x)^(n-1) / C(n,x)^n dx ), for n >= 1, where i^2 = -1.

%F (3.a) C(1,x) + i*S(x) = exp( i * Integral S'(x) / C(1,x) dx ).

%F (3.b) C(2,x)^2 + i*S(x)^2 = exp( 2*i * Integral S'(x)*S(x) / C(2,x)^2 dx ).

%F (3.c) C(3,x)^3 + i*S(x)^3 = exp( 3*i * Integral S'(x)*S(x)^2 / C(3,x)^3 dx ).

%F (3.d) C(4,x)^4 + i*S(x)^4 = exp( 4*i * Integral S'(x)*S(x)^3 / C(4,x)^4 dx ).

%F (4.a) C(n,x)^n = cos( n * Integral S'(x)*S(x)^(n-1) / C(n,x)^n dx ), for n >= 1.

%F (4.b) S(x)^n = sin( n * Integral S'(x)*S(x)^(n-1) / C(n,x)^n dx ), for n >= 1.

%F (5) C(n,x)^(2*n) - S(x)^(2*n) = cos( 2*n * Integral S'(x)*S(x)^(n-1) / C(n,x)^n dx ), for n >= 1.

%F (5.a) 2*C(n,x)^(2*n) = 1 + cos( 2*n * Integral S'(x)*S(x)^(n-1) / C(n,x)^n dx ), for n >= 1.

%F (5.b) 2*S(x)^(2*n) = 1 - cos( 2*n * Integral S'(x)*S(x)^(n-1) / C(n,x)^n dx ), for n >= 1.

%F (6) S(x) = Series_Reversion( Integral Product_{n>=1} (1 - x^(2*n))^(-(2*n-1)/(2*n)) dx ).

%e E.g.f.: 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! + 1142264265564479*x^19/19! + ...

%e such that

%e S(x) = Series_Reversion( Integral 1/((1 - x^2)^(1/2) * (1 - x^4)^(3/4) * (1 - x^6)^(5/6) * (1 - x^8)^(7/8) * ... * (1 - x^(2*n))^((2*n-1)/(2*n)) * ...) dx )

%e also,

%e S(x) = Integral C(1,x) * C(2,x)^3 * C(3,x)^5 * C(4,x)^7 * ... * C(n,x)^(2*n-1) * ... dx,

%e where

%e C(1,x) = (1 - S(x)^2)^(1/2) = 1 - x^2/2! + x^4/4! + 107*x^6/6! + 913*x^8/8! - 131449*x^10/10! - 46887791*x^12/12! + 4109309363*x^14/14! + ...

%e C(2,x) = (1 - S(x)^4)^(1/4) = 1 - 24*x^4/4! + 480*x^6/6! + 16128*x^8/8! - 1355520*x^10/10! - 78076416*x^12/12! - 14305374720*x^14/14! + ...

%e C(3,x) = (1 - S(x)^6)^(1/6) = 1 - 1080*x^6/6! + 60480*x^8/8! + 2358720*x^10/10! - 379969920*x^12/12! - 85742616960*x^14/14! + ...

%e C(4,x) = (1 - S(x)^8)^(1/8) = 1 - 80640*x^8/8! + 9676800*x^10/10! + 340623360*x^12/12! - 223250227200*x^14/14! + ...

%e ...

%e C(2,x)^3 = 1 - 72*x^4/4! + 1440*x^6/6! + 169344*x^8/8! - 18581760*x^10/10! - 1224165888*x^12/12! + 466315799040*x^14/14! + ...

%e C(3,x)^5 = 1 - 5400*x^6/6! + 302400*x^8/8! + 11793600*x^10/10! + 8877686400*x^12/12! - 4351736188800*x^14/14! + ...

%e C(4,x)^7 = 1 - 564480*x^8/8! + 67737600*x^10/10! + 2384363520*x^12/12! - 1562751590400*x^14/14! + ...

%e ...

%e SERIES REVERSION.

%e Let R(x) be the series reversion of S(x) so that S(R(x)) = x, then

%e R(x) = Integral Product_{n>=1} 1/(1 - x^(2*n))^((2*n-1)/(2*n)) dx

%e where

%e R(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! + 72527385885379425*x^19/19! + ... + A357228(n)*x^(2*n-1)/(2*n-1)! + ...

%o (PARI) /* Using Series Reversion of Integral of infinite product */

%o {a(n) = my(S); S = serreverse( intformal( prod(k=1, n, (1 - x^(2*k) + O(x^(2*n)) )^(-(2*k-1)/(2*k)) ) ) ); (2*n-1)! * polcoeff(S, 2*n-1)}

%o for(n=1, 20, print1(a(n), ", "))

%o (PARI) /* Using Integral of infinite product */

%o {a(n) = my(S=x); for(i=1, n,

%o S = intformal( prod(k=1, n, (1 - S^(2*k) + O(x^(2*n)))^((2*k-1)/(2*k)) ) ) ); (2*n-1)! * polcoeff(S, 2*n-1)}

%o for(n=1, 20, print1(a(n), ", "))

%Y Cf. A357228, A357551 (C(1,x)), A357230.

%K sign

%O 1,3

%A _Paul D. Hanna_, Oct 02 2022