OFFSET
0,4
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..300
FORMULA
G.f. C(x) = Sum_{n>=0} a(n) * x^(2*n)/(2*n)! and related odd function S(x) satisfies the following relations.
(1) S(x) = Integral Product_{n>=1} (1 + S(x)^(2*n))^((2*n-1)/(2*n)) dx.
(2) C(x) = 1 + Integral S(x) * Product_{n>=2} (1 + S(x)^(2*n))^((2*n-1)/(2*n)).
(3) C(x) = sqrt(1 + S(x)^2).
(4) C'(x)*C(x) = S'(x)*S(x).
(5) C(x) + S(x) = exp( Integral S'(x) / C(x) dx ).
(5.a) C(x) = cosh( Integral S'(x) / C(x) dx ).
(5.b) S(x) = sinh( Integral S'(x) / C(x) dx ).
(6) C(x)^2 + S(x)^2 = cosh( 2 * Integral S'(x) / C(x) dx ).
(6.a) 2*C(x)^2 = 1 + cosh( 2 * Integral S'(x) / C(x) dx ).
(6.b) 2*S(x)^2 = -1 + cosh( 2 * Integral S'(x) / C(x) dx ).
(7) S(x) = Series_Reversion( Integral Product_{n>=1} (1 + x^(2*n))^(-(2*n-1)/(2*n)) dx ).
EXAMPLE
G.f.: C(x) = 1 + x^2/2! + x^4/4! + 109*x^6/6! + 8689*x^8/8! + 1053481*x^10/10! + 243813361*x^12/12! + 75186825109*x^14/14! + 31749087943969*x^16/16! + ...
where C(x) = sqrt(1 + S^2) and
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(S); S = serreverse( intformal( prod(k=1, n+1, (1 + x^(2*k) + x*O(x^(2*n)) )^(-(2*k-1)/(2*k)) ) ) ); (2*n)! * polcoeff( sqrt(1 + S^2), 2*n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = my(S=x); for(i=1, n, S = intformal( prod(k=1, n+1, (1 + S^(2*k) + x*O(x^(2*n)))^((2*k-1)/(2*k)) ) ) ); (2*n)! * polcoeff( sqrt(1 + S^2), 2*n)}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 04 2022
STATUS
approved