A053984 a(n) = (2*n-1)*a(n-1) - a(n-2), a(0) = 0, a(1) = 1.

0, 1, 3, 14, 95, 841, 9156, 118187, 1763649, 29863846, 565649425, 11848774079, 271956154392, 6787055085721, 182978531160075, 5299590348556454, 164104322274089999, 5410143044696413513, 189190902242100382956, 6994653239913017755859, 272602285454365592095545

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