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A036244 Denominator of continued fraction given by C(n) = [ 1; 3, 5, 7, ...(2n-1)]. 4
1, 3, 16, 115, 1051, 11676, 152839, 2304261, 39325276, 749484505, 15778499881, 363654981768, 9107153044081, 246256787171955, 7150553981030776, 221913430199126011, 7330293750552189139, 256782194699525745876, 9508271497633004786551, 371079370602386712421365 (list; graph; refs; listen; history; text; internal format)
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
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
Numerators are sequence A025164. A058798.
Sequence in context: A159606 A211210 A177402 * A011818 A036248 A111555
KEYWORD
nonn,easy,frac
AUTHOR
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

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Last modified March 1 19:16 EST 2024. Contains 370443 sequences. (Running on oeis4.)