A194957 E.g.f. satisfies: A(x) = cos(x) + x*A(x)^2.

1, 1, 3, 24, 265, 3880, 71099, 1568896, 40538065, 1201214592, 40167734579, 1496537977088, 61482321148249, 2761475028475904, 134626530031186283, 7080317004476928000, 399585298475488156321, 24087998749960900673536, 1544790381323645227703651

Offset

0,3

Comments

The radius of convergence r of the e.g.f. A(x) satisfies: r = 1/(4*cos(r)) = limit (n+1)*a(n)/a(n+1) = 0.2585985822541... with A(r) = 1/(2*r) = 1.933498612565961...

Links

G. C. Greubel, Table of n, a(n) for n = 0..365

Formula

E.g.f.: A(x) = (1 - sqrt(1 - 4*x*cos(x))) / (2*x).

a(n) = (-1)^floor(n/2)*(1 + (-1)^n)/2 + n*Sum_{k=0..n-1} binomial(n-1,k)*a(k)*a(n-1-k) for n >= 0.

a(n) ~ n! * sqrt(cos(r)-sin(r)*r)/(2*sqrt(Pi)*r^(n+1/2)*n^(3/2)), where r = 0.258598582254189... is defined in the comment. - Vaclav Kotesovec, Sep 21 2013

Example

E.g.f.: A(x) = 1 + x + 3*x^2/2! + 24*x^3/3! + 265*x^4/4! + 3880*x^5/5! + ...
Related expansion:
A(x)^2 = 1 + 2*x + 8*x^2/2! + 66*x^3/3! + 776*x^4/4! + 11850*x^5/5! + ...

Mathematica

CoefficientList[Series[(1-Sqrt[1-4*x*Cos[x]])/(2*x), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Sep 21 2013 *)

Prog

(PARI) {a(n)=n!*polcoeff((1 - sqrt(1 - 4*x*cos(x +O(x^(n+2))))) / (2*x), n)}
(PARI) {a(n)=(-1)^(n\2)*(1 + (-1)^n)/2 +n*sum(k=0, n-1, binomial(n-1, k)*a(k)*a(n-1-k))}

Crossrefs

Cf. A194958, A194471.

Sequence in context: A218301 A233833 A219536 * A081133 A352278 A218223

Adjacent sequences: A194954 A194955 A194956 * A194958 A194959 A194960

Keyword

nonn

Author

Paul D. Hanna, Sep 06 2011

Status

approved