A177698 Expansion of e.g.f.: sin(arctan(x)).

0, 1, 0, -3, 0, 45, 0, -1575, 0, 99225, 0, -9823275, 0, 1404728325, 0, -273922023375, 0, 69850115960625, 0, -22561587455281875, 0, 9002073394657468125, 0, -4348001449619557104375, 0, 2500100833531245335015625, 0, -1687568062633590601135546875

Offset

0,4

Comments

Except periodic zeros for n even, and negative signs for n == 3 (mod 4), we find the same sequence as A079484 (determinant of M(2n-1) where M(k) is the k X k matrix with m(i,j)=j if i+j=k, m(i,j)=i otherwise).

References

L. Comtet and M. Fiolet, Sur les dérivées successives d'une fonction implicite. C. R. Acad. Sci. Paris Ser. A 278 (1974), 249-251.

Links

G. C. Greubel, Table of n, a(n) for n = 0..450

Formula

E.g.f.: sin(arctan(x)) = x/sqrt(1+x^2).

E.g.f.: x/(G(0)+x) where G(k)= 1 - 2*x/(1 + 1/G(k+1) ); (recursively defined continued fraction). - Sergei N. Gladkovskii, Dec 08 2012

a(n) ~ 2*sin(Pi*n/2)*n^n/exp(n). - Vaclav Kotesovec, Sep 25 2013

From Benedict W. J. Irwin, Nov 03 2016: (Start)

a(n) = y(n,n), n>0, where y(m+1,n) = (n-2*m)*y(m,n), with y(0,n)=1, for all n.

a(n) = Real part of -i*(2*i)^n*Gamma(1 + n/2)*Gamma(n/2)/Pi. (End)

From Michael Somos, May 04 2017: (Start)

a(n) = -n * (n-2) * a(n-2) for all n in Z.

a(n) = 1 / a(-n) for all odd n in Z.

a(n) = n! * binomial(n-1,(n-1)/2) * 2^(1-n) * (-1)^((n-1)/2) if n is odd > 0.

a(2*n + 1) = (-1)^n * A079484(n). (End)

Example

G.f. = x - 3*x^3 + 45*x^5 - 1575*x^7 + 99225*x^9 - 9823275*x^11 + ...
d^3y/dx^3 = 18/(1+x^2)^(5/2)*x^2 -3/(1+x^2)^(3/2) -15*x^4/(1+x^2)^(7/2).
For x = 0, we obtain a(3) = 0 - 3 + 0 = -3.

Maple

a:= n-> n! * coeff(series(sin(arctan(x)), x, n+1), x, n):
seq(a(n), n=0..30);

Mathematica

Table[n!*SeriesCoefficient[x/Sqrt[1+x^2], {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Sep 25 2013 *)
With[{nn=30}, CoefficientList[Series[Sin[ArcTan[x]], {x, 0, nn}], x] Range[ 0, nn-1]!] (* Harvey P. Dale, Nov 30 2015 *)
Join[{0}, Table[DifferenceRoot[Function[{y, m}, {y[1 + m] == (n - 2 m)*y[m], y[0] == 1}]][n], {n, 1, 20}]] (* Benedict W. J. Irwin, Nov 03 2016 *)
Join[{0}, Table[Re[-((I (2 I)^n Gamma[1 + n/2] Gamma[n/2])/Pi)], {n, 1, 20}]] (* Benedict W. J. Irwin, Nov 03 2016 *)
a[ n_] := If[ EvenQ[n], 0, I^(n - 1) n!! (n - 2)!!]; (* Michael Somos, May 04 2017 *)

Prog

(PARI) {a(n) = if( n%2==0, 0, n<0, 1 / self()(-n), n! * binomial(n-1, n\2) * 2^(1-n) * (-1)^(n\2))}; /* Michael Somos, May 04 2017 */
(PARI) x='x+O('x^50); concat([0], Vec(x/sqrt(1+x^2))) \\ G. C. Greubel, Sep 25 2018
(Magma) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); [0] cat Coefficients(R!(x/Sqrt(1+x^2))); // G. C. Greubel, Sep 25 2018

Crossrefs

Cf. A079484.

Sequence in context: A172396 A164806 A105751 * A009786 A012738 A193410

Adjacent sequences: A177695 A177696 A177697 * A177699 A177700 A177701

Keyword

sign

Author

Michel Lagneau, May 11 2010

Status

approved