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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 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

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