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A036244
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Denominator of continued fraction given by C(n) = [ 1; 3, 5, 7, ...(2n-1)].
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4
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1, 3, 16, 115, 1051, 11676, 152839, 2304261, 39325276, 749484505, 15778499881, 363654981768, 9107153044081, 246256787171955, 7150553981030776, 221913430199126011, 7330293750552189139, 256782194699525745876, 9508271497633004786551, 371079370602386712421365
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OFFSET
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1,2
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COMMENTS
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Denominators of convergents to coth(1) = 1.313035... = A073747.
Convergents: 1/1, 4/3, 21/16, 151/115, ... - Michael Somos, Sep 27 2017
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LINKS
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FORMULA
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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) = 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
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EXAMPLE
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G.f. = x + 3*x^2 + 16*x^3 + 115*x^4 + 1051*x^5 + 11676*x^6 + 152839*x^7 + ...
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MAPLE
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seq(denom(numtheory:-cfrac([seq(2*i-1, i=1..n)])), n=1..50); # Robert Israel, Apr 19 2015
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MATHEMATICA
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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 *)
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PROG
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(Sage)
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)
(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
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CROSSREFS
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KEYWORD
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nonn,easy,frac
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AUTHOR
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EXTENSIONS
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More terms from Larry Reeves (larryr(AT)acm.org), May 15 2001
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STATUS
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approved
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