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A195135 E.g.f. A(x) satisfies: A(x) = cos(x/A(x)) + sin(x/A(x)). 0
1, 1, -3, 20, -235, 3856, -81487, 2105536, -64314423, 2267105536, -90581348891, 4045219841024, -199679678243747, 10795704486465536, -634443738868419495, 40268829506038513664, -2745282759717516526831, 200066916824905348612096, -15521045994692397205950259, 1277100761225714616266653696 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Table of n, a(n) for n=0..19.

FORMULA

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

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

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

a(n) ~ (-1)^(n+1) * (s^2/sqrt(r^2+2*s^2)) * n^(n-1) / (exp(n) * r^n), where s = 0.528196651586997928322... is the root of the equation sin(2*s/sqrt(2-s^2)) = 1-s^2 and r = 0.2126668534407471004536... = s^2/sqrt(2-s^2). - Vaclav Kotesovec, Jan 11 2014

EXAMPLE

E.g.f.: A(x) = 1 + x - 3*x^2/2! + 20*x^3/3! - 235*x^4/4! + 3856*x^5/5! - 81487*x^6/6! +...

where

cos(x/A(x)) = 1 - x^2/2! + 6*x^3/3! - 71*x^4/4! + 1160*x^5/5! - 24481*x^6/6! +...

sin(x/A(x)) = x - 2*x^2/2! + 14*x^3/3! - 164*x^4/4! + 2696*x^5/5! - 57006*x^6/6! +...

SPECIAL VALUES.

A( Pi*sqrt(3+sqrt(5))/40 ) = sqrt(3+sqrt(5))/2 = 1.1441228056... where Pi*sqrt(3+sqrt(5))/40 = 0.17971839...

A( Pi/17*(cos(Pi/17) + sin(Pi/17)) ) = cos(Pi/17) + sin(Pi/17) = 1.16672261...

A( Pi/18*(cos(Pi/18) + sin(Pi/18)) ) = cos(Pi/18) + sin(Pi/18) = 1.15845593...

MATHEMATICA

CoefficientList[x/InverseSeries[Series[x*(Cos[x] + Sin[x]), {x, 0, 21}], x], x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 11 2014 *)

PROG

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

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

CROSSREFS

Sequence in context: A294603 A240957 A335871 * A201148 A248653 A218374

Adjacent sequences: A195132 A195133 A195134 * A195136 A195137 A195138

KEYWORD

sign

AUTHOR

Paul D. Hanna, Sep 09 2011

STATUS

approved

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Last modified February 9 01:59 EST 2023. Contains 360153 sequences. (Running on oeis4.)