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

1,2

Comments

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

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

Links

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

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

Example

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

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 *)

Prog

(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
(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

Crossrefs

Cf. A001040, A001053, A073747.

Numerators are sequence A025164. A058798.

Sequence in context: A159606 A211210 A177402 * A011818 A036248 A111555

Adjacent sequences: A036241 A036242 A036243 * A036245 A036246 A036247

Keyword

nonn,easy,frac

Author

Jeff Burch

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

Status

approved