OFFSET
0,3
COMMENTS
A related identity is sinh(x)/x = Product_{n>=1} cosh(x/2^n).
Motivated by the fixed point sr = Product_{n>=1} cosh(sr*(1/2)^n), where sr = sinh(sr)/sr = 1.3132837183534835944... (A133916).
The radius of convergence of e.g.f. A(x) is r = 0.536180023946684527...; A(x) evaluated at x = r satisfies (1) A(r) = 1 / [Sum_{n>=1} r^n * tanh(r^n * A(r))] and (2) A(r) = Product_{n>=1} cosh(r^n * A(r)), where A(r) = 1.726459616031644143... Note that r = 2/sqrt(d) where d is given in Vaclav Kotesovec's formula. - Paul D. Hanna, Mar 01 2024
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..201
FORMULA
E.g.f. A(x) = Sum_{n>=0} a(n) * x^(2*n)/(2*n)! satisfies the following formulas.
(1) A(x) = Product_{n>=1} cosh(x^n*A(x)).
(2) log(A(x)) = Sum_{n>=1} (-1)^(n-1)*A000182(n) * A(x)^(2*n) * x^(2*n)/((1-x^(2*n))*(2*n)!), where A000182 are the tangent numbers.
a(n) ~ c * d^n * (n-1)!^2, where d = 13.91357494878057309372... and c = 0.2556213529084458598... - Vaclav Kotesovec, Mar 01 2024
EXAMPLE
E.g.f.: A(x) = 1 + x^2/2! + 25*x^4/4! + 1801*x^6/6! + 251665*x^8/8! + 60023281*x^10/10! + 21783217897*x^12/12! + 11244708818617*x^14/14! + ...
where
A(x) = cosh(x*A(x)) * cosh(x^2*A(x)) * cosh(x^3*A(x)) * cosh(x^4*A(x)) * ...
RELATED SERIES.
log(A(x)) = x^2/2! + 22*x^4/4! + 1456*x^6/6! + 189232*x^8/8! + 43031296*x^10/10! + 15070050304*x^12/12! + 7582799641600*x^14/14! + ...
where the logarithm of A(x) may be written as
log(A(x)) = A(x)^2*x^2/((1-x^2)*2!) - 2*A(x)^4*x^4/((1-x^4)*4!) + 16*A(x)^6*x^6/((1-x^6)*6!) - 272*A(x)^8*x^8/((1-x^8)*8!) +- ...
in which the coefficients (A000182) are taken from the series for
log(cosh(x)) = x^2/2! - 2*x^4/4! + 16*x^6/6! - 272*x^8/8! + 7936*x^10/10! - 353792*x^12/12! + ... + (-1)^(n-1)*A000182(n)*x^(2*n)/(2*n)! + ...
SPECIFIC VALUES.
A(1/2) = sr = sinh(sr)/sr = 1.3132837183534835944... (A133916).
A(1/3) = 1.0732039012742053466040583737125980229145749599...
A(1/4) = 1.0360440144515230397124814819872716708635571534...
A(1/5) = 1.0218446382532843162980010372360052478618874316...
A(1/6) = 1.0147502698567245499165367962927287752921277594...
A(1/8) = 1.0080767433235165071886167844759835649060174961...
MATHEMATICA
nmax = 20; A[_] = 0; Do[A[x_] = Product[Cosh[x^k*A[x]], {k, 1, 2*nmax}] + O[x]^(2*nmax + 1) // Normal, 2*nmax + 1]; Table[(CoefficientList[A[x], x]*Range[0, 2*nmax]!)[[2*j-1]], {j, 1, nmax}] (* Vaclav Kotesovec, Mar 01 2024 *)
PROG
(PARI) {a(n) = my(A=1); for(m=1, n+1, A=truncate(A);
A = prod(k=1, m, cosh(x^k*A +O(x^(2*m+1))) ) ; ); (2*n)!*polcoeff(A, 2*n)}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 29 2024
STATUS
approved