OFFSET
1,2
COMMENTS
The e.g.f. is motivated by the following identities:
(1) F(x) = Integral cosh(x) / cos(F(x)) dx holds when F(x) = arcsin( sinh(x) ).
(2) F(x) = Integral sinh(2*x) / sin(2*F(x)) dx holds when F(x) = arcsin( sinh(x) ).
(3) F(x) = Integral sinh(F(x)) / sin(x) dx holds when F(x) = 2*arctanh( tan(x/2) ).
(4) F(x) = Integral sin(F(x)) / sinh(x) dx holds when F(x) = 2*arctan( tanh(x/2) ) = Integral 1/cosh(x) dx = Series_Reversion( Integral 1/cos(x) dx ).
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..200
FORMULA
E.g.f.: G(G(x)) where G(x) = log( (1 + sin(x))/cos(x) ) = arccosh( 1/cos(x) ) = arctanh(sin(x)) = arcsinh(tan(x)) = gd^(-1)(x), the inverse Gudermannian.
E.g.f. A(x) satisfies:
(1a) A(x) = Integral cosh(A(x)) / cos(x) dx.
(1b) B(x) = Integral cos(B(x)) / cosh(x) dx holds when A(B(x)) = x.
(2a) A(x) = Integral 1/(cos(x) * cos( arcsinh( tan(x) ) ) ) dx.
(2b) A(x) = Integral 1/(cos(x) * cos( arctanh( sin(x) ) ) ) dx.
(2c) A(x) = Integral 1/(cos(x) * cos( arccosh( 1/cos(x) ) ) ) dx.
(3a) A(x) = arccosh( 1/cos( arccosh( 1/cos(x) ) ) ).
(3b) A(x) = arccosh( 1/cos( arcsinh( tan(x) ) ) ).
(3c) A(x) = arccosh( 1/cos( arctanh( sin(x) ) ) ).
(3d) A(x) = arcsinh( tan( arccosh( 1/cos(x) ) ) ).
(3e) A(x) = arctanh( sin( arccosh( 1/cos(x) ) ) ).
(4a) A(x) = arctanh( sin( arcsinh( tan(x) ) ) ).
(4b) A(x) = arcsinh( tan( arctanh( sin(x) ) ) ).
(4c) A(x) = arcsinh( tan( arcsinh( tan(x) ) ) ).
(4d) A(x) = arctanh( sin( arctanh( sin(x) ) ) ).
From Vaclav Kotesovec, Oct 08 2017: (Start)
a(n) ~ sqrt(Pi)*2^(2*n) * n^(2*n-3/2) / ((arctan(sinh(Pi/2)))^(2*n-1) * exp(2*n)).
a(n) ~ n!^2 * 2^(2*n-1) / (sqrt(Pi) * n^(5/2) * (arctan(sinh(Pi/2)))^(2*n-1)).
(End)
EXAMPLE
E.g.f. A(x) = x + 2*x^3/3! + 20*x^5/5! + 472*x^7/7! + 20240*x^9/9! + 1375392*x^11/11! + 136036160*x^13/13! + 18472995712*x^15/15! + 3300092289280*x^17/17! + 750656264786432*x^19/19! + 211878817289753600*x^21/21! + ... + a(n)*x^(2*n-1)/(2*n-1)! + ...
such that A(x) = G(G(x)) where G(x) = log( (1 + sin(x))/cos(x) ).
RELATED SERIES.
Let G(x) = log( (1 + sin(x))/cos(x) ) = arccosh( 1/cos(x) ), then
G(x) = x + x^3/3! + 5*x^5/5! + 61*x^7/7! + 1385*x^9/9! + 50521*x^11/11! + 2702765*x^13/13! + 199360981*x^15/15! + ... + A000364(n-1)*x^(2*n-1)/(2*n-1)! + ...
where A(x) = G(G(x)).
The derivative of the e.g.f. A(x) is given by
A'(x) = cosh(A(x))/cos(x) = 1 + 2*x^2/2! + 20*x^4/4! + 472*x^6/6! + 20240*x^8/8! + 1375392*x^10/10! + ... + a(n)*x^(2*n-2)/(2*n-2)! + ...
Let B(x) be the series reversion of A(x), so that A(B(x)) = x, then
B(x) = Integral cos(B(x))/cosh(x) dx = x - 2*x^3/3! + 20*x^5/5! - 472*x^7/7! + 20240*x^9/9! - 1375392*x^11/11! + ... + (-1)^(n-1) * a(n)*x^(2*n-1)/(2*n-1)! + ...
E.g.f. A(x) as a series with coefficients a(n)/n! as reduced fractions begins:
A(x) = x + (1/3)*x^3 + (1/6)*x^5 + (59/630)*x^7 + (253/4536)*x^9 + (14327/415800)*x^11 + (32701/1496880)*x^13 + (144320279/10216206000)*x^15 + (151658653/16345929600)*x^17 + (1466125517161/237588086736000)*x^19 + (1182359471483/285105704083200)*x^21 + ...
MATHEMATICA
nmax = 20; Table[(CoefficientList[Series[1/(Cos[x] * Cos[Log[1/Cos[x] + Tan[x]]]), {x, 0, 2*nmax}], x] * Range[0, 2*nmax]!)[[2*k-1]], {k, 1, nmax}] (* Vaclav Kotesovec, Oct 08 2017 *)
PROG
(PARI) {a(n) = my(A=x, Ox=x*O(x^(2*n))); for(i=0, n, A = intformal( cosh(A +Ox) / cos(x +Ox))); (2*n-1)!*polcoeff(A, 2*n-1)}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 21 2017
STATUS
approved