A292806 - OEIS

%I #7 Sep 23 2017 21:40:40

%S 1,2,28,1048,75792,8997152,1589002688,390961266048,127846741426432,

%T 53632884600381952,28079728446200552448,17946985636126706997248,

%U 13752407157731907070595072,12445413772239663599454461952,13132326759927928089640745156608,15981710147225745975653754234830848,22219455702861159981173310810673446912,34999777375499159602747762386616587517952

%N E.g.f. A(x) satisfies: A(x) = Integral cosh(A(x)) / cos(A(x)) dx.

%H Paul D. Hanna, <a href="/A292806/b292806.txt">Table of n, a(n) for n = 1..150</a>

%F E.g.f.: Series_Reversion( Integral cos(x)/cosh(x) dx ).

%e E.g.f.: A(x) = x + 2*x^3/3! + 28*x^5/5! + 1048*x^7/7! + 75792*x^9/9! + 8997152*x^11/11! + 1589002688*x^13/13! + 390961266048*x^15/15! + 127846741426432*x^17/17! + 53632884600381952*x^19/19! +...

%e such that A'(x) = cosh(A(x)) / cos(A(x)).

%e RELATED SERIES.

%e Let B(x) be the series reversion of e.g.f. A(x), then

%e B(x) = x - 2*x^3/3! + 12*x^5/5! - 152*x^7/7! + 3472*x^9/9! - 126752*x^11/11! + 6781632*x^13/13! - 500231552*x^15/15! +...+ A000795(n-1)*x^(2*n-1)!/(2*n-1)! +...

%e then G'(x) = cos(x)/cosh(x).

%e Let G(x) be defined by G(G(x)) = A(x), then

%e G(x) = x + x^3/3! + 9*x^5/5! + 237*x^7/7! + 12385*x^9/9! + 1067225*x^11/11! + 136228105*x^13/13! + 24056468229*x^15/15! + 5614204466945*x^17/17! + 1677288189454257*x^19/19! + 626137638928559689*x^21/21! + 285873599602408829469*x^23/23! + 156375718123032150293473*x^25/25! +...

%o (PARI) {a(n) = my(A=x, Ox=x*O(x^(2*n))); for(i=0, n, A = intformal( cosh(A +Ox) / cos(A +Ox))); (2*n-1)!*polcoeff( G = A, 2*n-1)}

%o for(n=1,30,print1(a(n),", "))

%Y Cf. A292396, A000795.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Sep 23 2017