The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A053984 a(n) = (2*n-1)*a(n-1) - a(n-2), a(0) = 0, a(1) = 1. 13
 0, 1, 3, 14, 95, 841, 9156, 118187, 1763649, 29863846, 565649425, 11848774079, 271956154392, 6787055085721, 182978531160075, 5299590348556454, 164104322274089999, 5410143044696413513, 189190902242100382956, 6994653239913017755859, 272602285454365592095545 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Numerators of successive convergents to tan(1) using continued fraction 1/(1-1/(3-1/(5-1/(7-1/(9-1/(11-1/(13-1/15-...))))))). Equals eigensequence of an infinite lower triangular matrix with (1, 3, 5, 7,...) as the main diagonal and (0, -1, -1, -1,...) as the subdiagonal. - Gary W. Adamson, Apr 20 2009 LINKS G. C. Greubel, Table of n, a(n) for n = 0..400 S. Janson, A divergent generating function that can be summed and analysed analytically, Discrete Mathematics and Theoretical Computer Science; 2010, Vol. 12, No. 2, 1-22. FORMULA a(n) = (-1)^n*A053983(-1-n). - Michael Somos, Aug 23 2000 [See Somos's formula in A053983 which is valid for all n in Z.] E.g.f.: sin(1-sqrt(1-2*x))/sqrt(1-2*x). Cf. A036244. - Vladeta Jovovic, Aug 10 2006 Recurrence equation: a(n+1) = (2*n+1)*a(n) - a(n-1) with a(0) = 0 and a(1) = 1. a(n) = Sum_{k = 0..floor((n-1)/2)} (-1)^k*2^(n-2*k-1)*(n-2*k-1)!*binomial(n-k-1,k)*binomial(n-k-1/2,k+1/2), cf. A058798. - Peter Bala, Aug 01 2013 a(n) ~ sin(1)*2^(n+1/2)*n^n/exp(n). - Vaclav Kotesovec, Oct 05 2013 a(n) = (2*n-1)!!*hypergeometric([1 - n/2, 1/2 - n/2], [3/2, 1 - n, 1/2 - n], -1) for n >= 2. - Peter Luschny, Sep 10 2014 0 = a(n)*(+a(n+2)) + a(n+1)*(-a(n+1) + 2*a(n+2) - a(n+3)) + a(n+2)*(+a(n+2)) for all n in Z. - Michael Somos, Sep 11 2014 a(n) = SphericalBesselJ[n,1]*SphericalBesselY[0,1] - SphericalBesselJ[0,1]*SphericalBesselY[n,1]. - G. C. Greubel, May 10 2015 Sum_{n>=0} a(n-1)*t^n/n! = - cos(1 - sqrt(1-2*t)), where a(-1) = -1. - G. C. Greubel, May 10 2015 The SphericalBessel formula given by Greubel above can be rewritten as a(n) = sqrt(Pi/2)*(-cos(1)*BesselJ(n+1/2, 1)  + (-1)^n*sin(1)*BesselJ(-(n+1/2), 1)). - Wolfdieter Lang, Jun 14 2015 EXAMPLE a(10)=565649425 because 1/(1-1/(3-1/(5-1/(7-1/(9-1/(11-1/(13-1/(15-1/(17-1/19)))))))))=565649425/363199319. MAPLE f:= gfun:-rectoproc({a(n)=(2*n-1)*a(n-1)-a(n-2), a(0)=0, a(1)=1}, a(n), remember): map(f, [\$0..30]); # Robert Israel, May 14 2015 MATHEMATICA CoefficientList[Series[Sin[1-Sqrt[1-2*x]]/Sqrt[1-2*x], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 05 2013 *) RecurrenceTable[{a[n] == (2*n - 1)*a[n - 1] - a[n - 2], a[0] == 0,   a[1] == 1}, a, {n, 0, 50}] (* G. C. Greubel, Jan 22 2017 *) PROG (Sage) def A053984(n):     if n < 2: return n     return 2^n*gamma(n+1/2)*hypergeometric([1-n/2, 1/2-n/2], [3/2, 1 - n, 1/2 -n], -1) / sqrt(pi) [round(A053984(n).n(100)) for n in (0..20)] # Peter Luschny, Sep 10 2014 (PARI) a(n)={if(n<2, n, (2*n-1)*a(n-1)-a(n-2))} \\ Edward Jiang, Sep 10 2014 (PARI) {a(n) = my(a0, a1, s=n<0); if( abs(n) < 2, return(n)); if( n<0, n=-1-n); a0=s; a1=1; for(k=2, n, a2 = (2*k-1)*a1 - a0; a0=a1; a1=a2); (-1)^(s*n) * a1}; /* Michael Somos, Sep 11 2014 */ (MAGMA) [n le 2 select (n-1) else (2*n-3)*Self(n-1)-Self(n-2): n in [1..25] ]; // Vincenzo Librandi, May 12 2015 CROSSREFS Cf. A053983, A058798. Sequence in context: A094369 A005772 A233083 * A113181 A295105 A295106 Adjacent sequences:  A053981 A053982 A053983 * A053985 A053986 A053987 KEYWORD nonn,easy,frac AUTHOR Vladeta Jovovic, Apr 02 2000 EXTENSIONS Additional comments from Michael Somos, Aug 23 2000 More terms from Vladeta Jovovic, Aug 10 2006 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified August 3 13:21 EDT 2020. Contains 336198 sequences. (Running on oeis4.)