A201923 E.g.f. satisfies: A(x) = 1/(cos(x*A(x)) - sin(x*A(x))).

1, 1, 5, 44, 581, 10256, 227529, 6088256, 190930729, 6870227200, 279066777613, 12632667642880, 630670054092525, 34426087332253696, 2039903110075608977, 130404672744539242496, 8946117466489960168913, 655585000075494566199296, 51111210765059412626238741

Offset

0,3

Comments

Compare e.g.f. to: LambertW(-x)/(-x) = (1/x)*Series_Reversion(x*(cosh(x) - sinh(x))).

The radius of convergence r of e.g.f. A(x) is given by:

r = t*(cos(t) - sin(t)) where tan(t) = (1-t)/(1+t), which evaluates as:

r = 0.21266685344074710045360679397024815598865409988038...

t = 0.40262817418811160981993252391123072456350647779608...

Further, A(r) = 1/(cos(t) - sin(t)), thus

A(r) = 1.89323426605496483543109751303457163422769666683274...

Links

Table of n, a(n) for n=0..18.

Formula

E.g.f. satisfies: A( x*(cos(x) - sin(x)) ) = 1/(cos(x) - sin(x)).

E.g.f: (1/x) * Series_Reversion( x*(cos(x) - sin(x)) ).

a(n) = [x^n/n!] 1/(cos(x)-sin(x))^(n+1) / (n+1).

a(n) ~ n^(n-1) * sqrt((t*cos(2*t))/(3+sin(2*t))) / (exp(n) * r^(n+1)), where r and t were described above. - Vaclav Kotesovec, Jan 12 2014

Example

E.g.f.: A(x) = 1 + x + 5*x^2/2! + 44*x^3/3! + 581*x^4/4! + 10256*x^5/5! +...
where
1/(cos(x)-sin(x)) = 1 + x + 3*x^2/2! + 11*x^3/3! + 57*x^4/4! + 361*x^5/5! + 2763*x^6/6! +...+ A001586(n)*x^n/n! +...
The coefficient of x^n/n! in powers of G(x) = 1/(cos(x)-sin(x)) begins:
G^1: [(1), 1, 3, 11, 57, 361, 2763, 24611, ..., A001586(n), ...];
G^2: [1,(2), 8, 40, 256, 1952, 17408, 177280, ..., A000828(n+1), ...];
G^3: [1, 3,(15), 93, 705, 6243, 63375, 724413, ...];
G^4: [1, 4, 24,(176), 1536, 15424, 175104, 2214656, ...];
G^5: [1, 5, 35, 295,(2905), 32525, 407435, 5638495, ...];
G^6: [1, 6, 48, 456, 4992, (61536), 841728, 12633216, ...];
G^7: [1, 7, 63, 665, 8001, 107527, (1592703), 25738265, ...];
G^8: [1, 8, 80, 928, 12160, 176768, 2816000, (48706048), ...]; ...
where coefficients in parenthesis form the initial terms of this sequence:
[1/1, 2/2, 15/3, 176/4, 2905/5, 61536/6, 1592703/7, 48706048/8, ...].

Mathematica

CoefficientList[1/x*InverseSeries[Series[x*(Cos[x] - Sin[x]), {x, 0, 21}], x], x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 12 2014 *)

Prog

(PARI) {a(n)=local(X=x+x*O(x^n)); n!*polcoeff(1/x*serreverse(x*(cos(X)-sin(X) )), n)}
(PARI) {a(n)=local(A=1+x, X=x+x*O(x^n)); for(i=1, n, A=1/(cos(X*A) - sin(X*A))); n!*polcoeff(A, n)}

Crossrefs

Cf. A001586.

Sequence in context: A106273 A349836 A052803 * A222059 A336290 A214396

Adjacent sequences: A201920 A201921 A201922 * A201924 A201925 A201926

Keyword

nonn

Author

Paul D. Hanna, Dec 06 2011

Status

approved