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A053983
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a(n) = (2*n-1)*a(n-1) - a(n-2), a(0)=a(1)=1.
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9
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1, 1, 2, 9, 61, 540, 5879, 75887, 1132426, 19175355, 363199319, 7608010344, 174621038593, 4357917954481, 117489163732394, 3402827830284945, 105370173575100901, 3473812900148044788, 121478081331606466679, 4491215196369291222335, 175035914577070751204386
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OFFSET
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0,3
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COMMENTS
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Denominators of successive convergents to tan(1) using continued fraction 1/(1-1/(3-1/(5-1/(7-1/(9-1/(11-1/(13-1/15-...))))))). - Michael Somos, Aug 07 2000
Equals eigensequence of an infinite lower triangular matrix with (1, 3, 5, ...) as the main diagonal and (-1, -1, -1, ...) as the subdiagonal. - Gary W. Adamson, Apr 20 2009
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LINKS
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FORMULA
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E.g.f.: cos(1-sqrt(1-2*x))/sqrt(1-2*x). If a(0)=0, a(n)=0, 1, 1, 2, 9, 61, 540, 5879, 75887, 1132426, ... then e.g.f. = sin(1)*cos(sqrt(1-2*x))-cos(1)*sin(sqrt(1-2*x)). - Miklos Kristof, Jun 15 2005, corrected by Vaclav Kotesovec, Jul 31 2014
a(n) = Sum_{k=0..floor(n/2)} (-1)^k*2^(n-2*k)*(n-2*k)!*binomial(n-k,k) * binomial(n-k-1/2,k-1/2). Cf. A058798. - Peter Bala, Aug 01 2013
a(n) = 2^n*Gamma(n+1/2)*hypergeometric([1/2-n/2, -n/2], [1/2, 1/2-n, -n], -1)/sqrt(Pi) for n >= 2. - Peter Luschny, Sep 10 2014
0 = a(n)*(+a(n+2)) + a(n+1)*(-a(n+1) + 2*a(n+2) - a(n+3)) + a(n+2)*(+a(n+2)) for all n in Z. - Michael Somos, Sep 11 2014
(1/(2^n*n!)) * Integral_{x = 0..1} (1 - x^2)^n*cos(x) dx = a(n)*sin(1) - A053984(n)*cos(1). Hence A053984(n)/a(n) -> tan(1) as n -> infinity. - Peter Bala, Mar 06 2015
a(n) = SphericalBesselJ[0,1]*SphericalBesselJ[n,1] + SphericalBesselY[0,1]*SphericalBesselY[n,1]. - G. C. Greubel, May 10 2015
Sum_{n>0} a(n-1) t^n/n! = sin(1 - sqrt(1-2t)). - G. C. Greubel, May 10 2015
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EXAMPLE
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a(10) = 363199319 because 1/(1-1/(3-1/(5-1/(7-1/(9-1/(11-1/(13-1/(15-1/(17-1/19))))))))) = 565649425/363199319.
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MAPLE
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E(x):=sin(1)*cos(sqrt(1-2*x))-cos(1)*sin(sqrt(1-2*x)): f[0]:=E(x): for n from 1 to 30 do f[n]:=diff(f[n-1], x) od: x:=0: for n from 1 to 30 do f[n]:=simplify(f[n]/(sin(1)^2+cos(1)^2)) od: seq(f[n], n=1..30); # Miklos Kristof, Jun 15 2005
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MATHEMATICA
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RecurrenceTable[{a[0]==a[1]==1, a[n]==(2n-1)a[n-1]-a[n-2]}, a, {n, 20}] (* Harvey P. Dale, Dec 21 2011 *)
CoefficientList[Series[Cos[1-Sqrt[1-2*x]]/Sqrt[1-2*x], {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Jul 31 2014 *)
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PROG
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(Sage)
if n < 2: return 1
return 2^n*gamma(n+1/2)*hypergeometric([1/2-n/2, -n/2], [1/2, 1/2-n, -n], -1)/sqrt(pi)
(PARI) a(n)={if(n<2, 1, (2*n-1)*a(n-1)-a(n-2))} \\ Edward Jiang, Sep 10 2014
(PARI) {a(n) = my(a0, a1, s=n<0); if( n>-3 && n<1, return(n+1)); if( n<0, n=-1-n); a0=1-s; a1=1; for(k=2, n, a2 = (2*k-1)*a1 - a0; a0=a1; a1=a2); (-1)^(s*n) * a1}; /* Michael Somos, Sep 11 2014 */
(Magma) [1] cat [ n le 2 select n else (2*n-1)*Self(n-1)-Self(n-2): n in [1..25] ]; // Vincenzo Librandi, Mar 08 2015
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CROSSREFS
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KEYWORD
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easy,frac,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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