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A292396 E.g.f. A(x) satisfies: A(x) = Integral cosh(A(x)) / cos(x) dx. 2
1, 2, 20, 472, 20240, 1375392, 136036160, 18472995712, 3300092289280, 750656264786432, 211878817289753600, 72678286573542807552, 29779155737410909573120, 14365934044931988456579072, 8059896286109090587343011840, 5203589243375950233355757780992, 3830521688619915067535686289653760, 3189244099371608285093534127085453312, 2981890775446940839437012657918612602880 (list; graph; refs; listen; history; text; internal format)
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

Cf. A000364 (secant numbers).

Sequence in context: A009252 A210901 A274572 * A274738 A012816 A012340

Adjacent sequences:  A292393 A292394 A292395 * A292397 A292398 A292399

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Sep 21 2017

STATUS

approved

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Last modified December 8 20:37 EST 2021. Contains 349596 sequences. (Running on oeis4.)