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A190392 E.g.f. A(x) satisfies A'(x)=sin(A(x))+cos(A(x)). 3
1, 1, 0, -4, -12, 4, 240, 1184, -1008, -59504, -401280, 643136, 38584128, 323581504, -848090880, -51666451456, -509739310848, 2004840714496, 123888658698240, 1386061762251776, -7721141999864832, -483475390212586496, -5974101514137292800, 45231727252157947904 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Let f(x) be a smooth function. The autonomous differential equation A'(x) = f(A(x)), with initial condition A(0) = 0, is separable and the solution is given by A(x) = inverse function of int {t = 0..x} 1/f(t). The inversion of the integral int {t = 0..x} 1/f(t) is most conveniently found by applying [Dominici, Theorem 4.1]. The result is A(x) = sum {n>=1} D^(n-1)[f](0)*x^n/n!, where the nested derivative D^n[f](x)is defined recursively as D^0[f](x) = 1 and D^(n+1)[f](x) = d/dx(f(x)*D^n[f](x)) for n >= 0. See A145271 for the coefficients in the expansion of D^n[f](x) in powers of f(x). In the present case we take f(x) = sin(x)+cos(x). - Peter Bala, Aug 27 2011

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..200

D. Dominici, Nested derivatives: A simple method for computing series expansions of inverse functions. arXiv:math/0501052v2 [math.CA]

FORMULA

E.g.f.: A(x) = inverse of int {t = 0..x} 1/(sin(t)+cos(t)) = series reversion (x-x^2/2!+3*x^3/3!-11*x^4/4!+57*x^5/5!-...) = x+x^2/2!-4*x^4/4!-12*x^5/5!+.... a(n) = D^(n-1)[sin(x)+cos(x)](0), where the nested derivative operator D^n is defined above. Compare with A012244. -Peter Bala, Aug 27 2011

E.g.f.: A(x) = 2*arctan((sqrt(2)-1)*exp(sqrt(2)*x))-Pi/4. Compare with A028296. - Peter Bala, Sep 02 2011

G.f.: 1/G(0) where G(k) = 1 - 2*x*(k+1)/(1 + 1/(1 - 2*x*(k+1)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 10 2013.

G.f.: -1/x/Q(0), where Q(k)= 2*k+1 - 1/x + (k+1)*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Apr 15 2013

G.f.: T(0)/(1-x), where T(k) = 1 - x^2*(k+1)^2/( x^2*(k+1)^2 + (1-x-2*x*k)*(1-3*x-2*x*k)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 17 2013

E.g.f. if offset 0: 2^(1/2)/(2^(1/2)*cosh(x*2^(1/2))-sinh(x*2^(1/2))). - Sergei N. Gladkovskii, Nov 10 2013

a(n) ~ -(n-1)! * 2^(3*n/2+1) * sin(n*arctan(Pi/log(3 - 2*sqrt(2)))) / (Pi^2 + log(3 - 2*sqrt(2))^2)^(n/2). - Vaclav Kotesovec, Jan 07 2014

MAPLE

A := x ; for i from 1 to 35 do sin(A)+cos(A) ; convert(taylor(%, x=0, 25), polynom) ; A := int(%, x) ; print(A) ; end do:

for i from 1 to 25 do printf("%d, ", coeftayl(A, x=0, i)*i!) ; end do: # R. J. Mathar, Jun 03 2011

MATHEMATICA

max = 24; f[x_] := Sum[c[k]*x^k, {k, 0, max}]; c[0] = 0; c[1] = 1; se = Series[ f[x] - (2*ArcTan[(Sqrt[2] - 1)*Exp[Sqrt[2]*x]] - Pi/4), {x, 0, max}]; coes = CoefficientList[se, x] // Simplify; sol = Solve[Thread[coes == 0]]; (Table[c[k], {k, 0, max}] /. sol // First // Rest)*Range[max]! (* Jean-Fran├žois Alcover, Feb 21 2013, after Peter Bala *)

PROG

(Maxima)

g(n):=(-1)^floor(n/2)*1/n!;

a(n):=T190015_Solve(n, g);

CROSSREFS

Cf. A001586, A012244, A028296.

Sequence in context: A262621 A255289 A203031 * A133517 A258565 A145046

Adjacent sequences:  A190389 A190390 A190391 * A190393 A190394 A190395

KEYWORD

sign

AUTHOR

Vladimir Kruchinin, May 09 2011

STATUS

approved

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Last modified September 25 19:48 EDT 2017. Contains 292500 sequences.