A215638 E.g.f.: Series_Reversion( x/(cos(x) + sin(x)) ).

1, 2, 3, -16, -255, -1824, 875, 279040, 5019777, 34163200, -822579021, -35005820928, -598199789759, 1612679634944, 456446015789595, 15087057004396544, 168732823928006145, -8490541860738957312, -560665092239473776157, -14661549133472721797120

Offset

1,2

Comments

Lim sup n->infinity (|a(n)|/n!)^(1/n) = 1.991787... - Vaclav Kotesovec, Jan 22 2014

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..400

Formula

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

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

a(n) = n*A215639(n-1).

Example

E.g.f.: A(x) = x + 2*x^2/2! + 3*x^3/3! - 16*x^4/4! - 255*x^5/5! - 1824*x^6/6! +...
where A(x/(cos(x) + sin(x))) = x and A(x) = x*(cos(A(x)) + sin(A(x))).
Related expansions:
cos(A(x)) = 1 - x^2/2! - 6*x^3/3! - 23*x^4/4! + 40*x^5/5! + 2159*x^6/6! + 26656*x^7/7! + 114577*x^8/8! +...
sin(A(x)) = x + 2*x^2/2! + 2*x^3/3! - 28*x^4/4! - 344*x^5/5! - 2034*x^6/6! + 8224*x^7/7! + 443176*x^8/8! +...

Mathematica

Rest[CoefficientList[InverseSeries[Series[x/(Sqrt[2]*Cos[Pi/4-x]), {x, 0, 20}], x], x]*Range[0, 20]!] (* Vaclav Kotesovec, Jan 22 2014 *)

Prog

(PARI) {a(n)=local(X=x+x^2*O(x^n)); n!*polcoeff(serreverse(x/(cos(X)+sin(X))), n)}
(PARI) {a(n)=local(X=x+x^2*O(x^n)); n!*polcoeff(x*(cos(X)+sin(X))^n/n, n)}
(PARI) {a(n)=local(A=x+x^2*O(x^n)); for(i=1, n, A=x*(cos(A)+sin(A))); n!*polcoeff(A, n)}
for(n=1, 31, print1(a(n), ", "))

Crossrefs

Cf. A215368, A215639.

Sequence in context: A141309 A179442 A299424 * A057997 A290590 A263802

Adjacent sequences: A215635 A215636 A215637 * A215639 A215640 A215641

Keyword

sign

Author

Paul D. Hanna, Aug 18 2012

Status

approved