A195134 E.g.f. A(x) satisfies: A(x) = cosh(x*A(x)) + sin(x*A(x)).

1, 1, 3, 14, 93, 796, 8407, 105832, 1551865, 26033680, 492708491, 10400139232, 242507271061, 6195709678016, 172208913873375, 5175087678675584, 167222667351260145, 5781987852483789056, 213003988054590430099, 8328278686225469009408, 344418854322690984069581

Offset

0,3

Comments

Compare e.g.f. to W(x) = LambertW(-x)/(-x) = Sum_{n>=0} (n+1)^(n-1)*x^n/n!, which satisfies: W(x) = cosh(x*W(x)) + sinh(x*W(x)).

Links

Table of n, a(n) for n=0..20.

Formula

E.g.f. A(x) satisfies: A( x/(cosh(x) + sin(x)) ) = cosh(x) + sin(x).

E.g.f.: A(x) = (1/x)*Series_Reversion( x/(cosh(x) + sin(x)) ).

a(n) = [x^n] (cosh(x) + sin(x))^(n+1)/(n+1), where [x^n] F(x) denotes the coefficient of x^n in F(x).

a(n) ~ sqrt(t/(cosh(t)-sin(t))) * n^(n-1) * (cos(t)+sinh(t))^(n+3/2) / exp(n), where t = 1.523907509249588... is the root of the equation t*(cos(t) + sinh(t)) = cosh(t) + sin(t). - Vaclav Kotesovec, Jan 11 2014

Example

E.g.f.: A(x) = 1 + x + 3*x^2/2! + 14*x^3/3! + 93*x^4/4! + 796*x^5/5! +...
where
cosh(x*A(x)) = 1 + x^2/2! + 6*x^3/3! + 49*x^4/4! + 480*x^5/5! + 5641*x^6/6! +...
sin(x*A(x)) = x + 2*x^2/2! + 8*x^3/3! + 44*x^4/4! + 316*x^5/5! + 2766*x^6/6! +...

Mathematica

CoefficientList[1/x*InverseSeries[Series[x/(Cosh[x] + Sin[x]), {x, 0, 20}], x], x] * Range[0, 19]! (* Vaclav Kotesovec, Jan 11 2014 *)

Prog

(PARI) {a(n)=local(X=x+x*O(x^n)); n!*polcoeff(1/x*serreverse(x/((cosh(X) + sin(X)))), n)}
(PARI) {a(n)=local(X=x+x*O(x^(2*n))); n!*polcoeff((cosh(X)+sin(X))^(n+1)/(n+1), n)}

Crossrefs

Sequence in context: A259903 A101220 A078456 * A089462 A088342 A358118

Adjacent sequences: A195131 A195132 A195133 * A195135 A195136 A195137

Keyword

nonn

Author

Paul D. Hanna, Sep 09 2011

Status

approved