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%I #55 Feb 05 2022 16:42:19
%S 1,1,2,9,61,540,5879,75887,1132426,19175355,363199319,7608010344,
%T 174621038593,4357917954481,117489163732394,3402827830284945,
%U 105370173575100901,3473812900148044788,121478081331606466679,4491215196369291222335,175035914577070751204386
%N a(n) = (2*n-1)*a(n-1) - a(n-2), a(0)=a(1)=1.
%C 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
%C 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
%H G. C. Greubel, <a href="/A053983/b053983.txt">Table of n, a(n) for n = 0..400</a>
%F a(n) = -(-1)^n*A053984(-1-n). - _Michael Somos_, Aug 07 2000
%F 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
%F 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
%F a(n) ~ cos(1) * 2^(n+1/2) * n^n / exp(n). - _Vaclav Kotesovec_, Jul 31 2014
%F 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
%F 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
%F (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
%F a(n) = SphericalBesselJ[0,1]*SphericalBesselJ[n,1] + SphericalBesselY[0,1]*SphericalBesselY[n,1]. - _G. C. Greubel_, May 10 2015
%F Sum_{n>0} a(n-1) t^n/n! = sin(1 - sqrt(1-2t)). - _G. C. Greubel_, May 10 2015
%e 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.
%p 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
%t 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 *)
%t CoefficientList[Series[Cos[1-Sqrt[1-2*x]]/Sqrt[1-2*x], {x, 0, 20}], x] * Range[0, 20]! (* _Vaclav Kotesovec_, Jul 31 2014 *)
%o (Sage)
%o def A053983(n):
%o if n < 2: return 1
%o return 2^n*gamma(n+1/2)*hypergeometric([1/2-n/2, -n/2], [1/2, 1/2-n, -n], -1)/sqrt(pi)
%o [round(A053983(n).n(100)) for n in (0..20)] # _Peter Luschny_, Sep 10 2014
%o (PARI) a(n)={if(n<2,1,(2*n-1)*a(n-1)-a(n-2))} \\ _Edward Jiang_, Sep 10 2014
%o (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 */
%o (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
%Y Cf. A053984, A058798.
%K easy,frac,nonn
%O 0,3
%A _Vladeta Jovovic_, Apr 02 2000
%E Additional comments from _Michael Somos_, Aug 23 2000
%E More terms from _Miklos Kristof_, Jun 15 2005
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