A036244 Denominator of continued fraction given by C(n) = [ 1; 3, 5, 7, ...(2n-1)].

1, 3, 16, 115, 1051, 11676, 152839, 2304261, 39325276, 749484505, 15778499881, 363654981768, 9107153044081, 246256787171955, 7150553981030776, 221913430199126011, 7330293750552189139, 256782194699525745876, 9508271497633004786551, 371079370602386712421365

Offset: 1

Author

Jeff Burch

Keywords

• nonn
• easy
• frac

Comments

Denominators of convergents to coth(1) = 1.313035... = A073747.

Convergents: 1/1, 4/3, 21/16, 151/115, ... - Michael Somos, Sep 27 2017

Examples

			G.f. = x + 3*x^2 + 16*x^3 + 115*x^4 + 1051*x^5 + 11676*x^6 + 152839*x^7 + ...
		

Links

• Robert Israel, Table of n, a(n) for n = 1..369

Crossrefs

Cf. A001040, A001053, A073747.

Numerators are sequence A025164. A058798.

Programs

Magma

I:=[1,3]; [n le 2 select I[n] else (2*n-1)*Self(n-1)+Self(n-2): n in [1..30]]; // Vincenzo Librandi, Apr 19 2015

Maple

seq(denom(numtheory:-cfrac([seq(2*i-1,i=1..n)])),n=1..50); # Robert Israel, Apr 19 2015

Mathematica

Rest[CoefficientList[Series[(E^(1-(1-2*x)^(1/2))/2 - E^(-1+(1-2*x)^(1/2))/2) / (1-2*x)^(1/2), {x, 0, 20}], x]* Range[0, 20]!] (* Vaclav Kotesovec, Oct 05 2013 *)
a[ n_ ] := a[n] =a[n-2]+(2 n-1) a[n-1]; a[0] := 0; a[1] := 1.  RecurrenceTable[{a[0]==0, a[1]==1, a[n]==a[n-2]+(2n-1)a[n-1]}, a, {n, 20}] (* G. C. Greubel,Apr 23 2015 *)
a[ n_] := (BesselK[ 1/2, 1] BesselI[ n + 1/2, -1] - BesselI[ 1/2, -1] BesselK[n + 1/2, 1])  I // FunctionExpand // Simplify; (* Michael Somos, Sep 27 2017 *)
Table[FromContinuedFraction[Range[1,2n+1,2]],{n,0,20}]//Denominator (* Harvey P. Dale, May 06 2018 *)
Convergents[Coth[1], 20] // Denominator (* Jean-François Alcover, Jun 15 2019 *)

Sage

def A036244(n):
    if n == 1: return 1
    return 2^n*gamma(n+1/2)*hypergeometric([1/2-n/2, 1-n/2], [3/2, 1/2-n, 1-n], 1)/sqrt(pi)
[round(A036244(n).n(100)) for n in (1..20)] # Peter Luschny, Sep 11 2014

Formula

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

E.g.f.: sinh(1-(1-2*x)^(1/2))/(1-2*x)^(1/2). - Vladeta Jovovic, Jan 30 2004

E.g.f.: cosh(1-(1-2*x)^(1/2))/(1-2*x) + sinh(1-(1-2*x)^(1/2))/((1-2*x)^(3/2)).

E.g.f. G(0)/(1-2*x) where G(k)= 1 + 2*x/((2*k+1)*(1-2*x+sqrt(1-2*x))+(2*k+1)*(4*x^2-2*x)/(-1+2*x+sqrt(1-2*x) + (2*k+2)/G(k+1))); (continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Jul 01 2012

a(n) = Sum_{k=0..floor((n-1)/2)} 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) ~ (exp(2)-1)*2^(n-1/2)*n^n/exp(n+1). - Vaclav Kotesovec, Oct 05 2013

a(n) = A001147(n)*hypergeometric([1/2-n/2, 1-n/2], [3/2, 1/2-n, 1-n], 1) for n >= 2. - Peter Luschny, Sep 11 2014

a(n) = i*(BesselK[1/2,1]*BesselI[n+1/2,-1] - BesselI[1/2,-1]*BesselK[n+1/2,1]) for n>=0 (where a(0) = 0). - G. C. Greubel, Apr 18 2015

a(n) = A025164(-1-n) for all n in Z. - Michael Somos, Sep 27 2017

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), May 15 2001

More terms from Benoit Cloitre, Dec 20 2002

More terms from Vladeta Jovovic, Jan 30 2004