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A290879 E.g.f. L(x) of aerated sequence satisfies: L(x) = Integral 1 / sqrt( cosh(2*L(x)) ) dx. 5
1, -2, 44, -2840, 367760, -79719200, 26016555200, -11921650083200, 7300922254496000, -5759532173685056000, 5688335131502291840000, -6875441991877610827520000, 9983390897443366347676160000, -17148197258942716354314368000000, 34392433372153876998446324480000000, -79646680171456811546338888517120000000, 210930739548407111241584046599398400000000 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Robert Israel, Table of n, a(n) for n = 1..200

FORMULA

E.g.f.: L(x) = Series_Reversion( Integral sqrt( cosh(2*x) ) dx ).

Let C(x) and S(x) be the e.g.f.s of A290880 and A290881, respectively, then e.g.f. L(x) satisfies:

(1) L(x) = log(C(x) + S(x)),

(2) cosh(2*L(x)) = C(x)^2 + S(x)^2,

(3) cosh(L(x)) = C(x) and sinh(L(x)) = S(x),

where C(x)^2 - S(x)^2 = 1 and C'(x)^2 + S'(x)^2 = 1.

EXAMPLE

E.g.f.: L(x) = x - 2*x^3/3! + 44*x^5/5! - 2840*x^7/7! + 367760*x^9/9! - 79719200*x^11/11! + 26016555200*x^13/13! - 11921650083200*x^15/15! + 7300922254496000*x^17/17! - 5759532173685056000*x^19/19! + 5688335131502291840000*x^21/21! +...

where

1/sqrt( cosh(2*L(x)) ) = 1 - 2*x^2/2! + 44*x^4/4! - 2840*x^6/6! + 367760*x^8/8! - 79719200*x^10/10! + 26016555200*x^12/12! - 11921650083200*x^14/14! +...

MAPLE

Q:= series(Int(sqrt(cosh(2*t)), t), t, 100):

S:= series(RootOf(Q-y, t), y, 100):

seq(coeff(S, y, j)*j!, j=1..100, 2); # Robert Israel, Aug 13 2017

MATHEMATICA

terms = 17; m = 2 terms; cc = CoefficientList[InverseSeries[Integrate[Sqrt[ Cosh[2 x]] + O[x]^m, x], x], x]; DeleteCases[cc * Range[0, m-1]!, 0] (* Jean-Fran├žois Alcover, Apr 02 2019 *)

PROG

(PARI) {a(n) = my(L=x); for(i=1, n, L = intformal( 1/sqrt(cosh(2*L + O(x^(2*n+2)))) )); (2*n-1)!*polcoeff(L, 2*n-1)}

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

(PARI) {a(n) = my(L=x); L = serreverse( intformal( sqrt(cosh(2*x + O(x^(2*n+2)))) )); (2*n-1)!*polcoeff(L, 2*n-1)}

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

CROSSREFS

Cf. A290880, A290881, A290882, A290883, A153302.

Sequence in context: A267070 A054732 A161722 * A054914 A329021 A275307

Adjacent sequences:  A290876 A290877 A290878 * A290880 A290881 A290882

KEYWORD

sign

AUTHOR

Paul D. Hanna, Aug 13 2017

STATUS

approved

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Last modified July 13 04:31 EDT 2020. Contains 335673 sequences. (Running on oeis4.)