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A290879 E.g.f. L(x) of aerated sequence satisfies: L(x) = Integral 1 / sqrt( cosh(2*L(x)) ) dx.

%I A290879 #16 Apr 02 2019 11:48:56
%S A290879 1,-2,44,-2840,367760,-79719200,26016555200,-11921650083200,
%T A290879 7300922254496000,-5759532173685056000,5688335131502291840000,
%U A290879 -6875441991877610827520000,9983390897443366347676160000,-17148197258942716354314368000000,34392433372153876998446324480000000,-79646680171456811546338888517120000000,210930739548407111241584046599398400000000
%N A290879 E.g.f. L(x) of aerated sequence satisfies: L(x) = Integral 1 / sqrt( cosh(2*L(x)) ) dx.
%H A290879 Robert Israel, <a href="/A290879/b290879.txt">Table of n, a(n) for n = 1..200</a>
%F A290879 E.g.f.: L(x) = Series_Reversion( Integral sqrt( cosh(2*x) ) dx ).
%F A290879 Let C(x) and S(x) be the e.g.f.s of A290880 and A290881, respectively, then e.g.f. L(x) satisfies:
%F A290879 (1) L(x) = log(C(x) + S(x)),
%F A290879 (2) cosh(2*L(x)) = C(x)^2 + S(x)^2,
%F A290879 (3) cosh(L(x)) = C(x) and sinh(L(x)) = S(x),
%F A290879 where C(x)^2 - S(x)^2 = 1 and C'(x)^2 + S'(x)^2 = 1.
%e A290879 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! +...
%e A290879 where
%e A290879 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! +...
%p A290879 Q:= series(Int(sqrt(cosh(2*t)),t),t,100):
%p A290879 S:= series(RootOf(Q-y,t),y,100):
%p A290879 seq(coeff(S,y,j)*j!,j=1..100,2); # _Robert Israel_, Aug 13 2017
%t A290879 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-François Alcover_, Apr 02 2019 *)
%o A290879 (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)}
%o A290879 for(n=1,20, print1(a(n),", "))
%o A290879 (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)}
%o A290879 for(n=1,20, print1(a(n),", "))
%Y A290879 Cf. A290880, A290881, A290882, A290883, A153302.
%K A290879 sign
%O A290879 1,2
%A A290879 _Paul D. Hanna_, Aug 13 2017