A093985 - OEIS

%I #28 Feb 14 2024 05:45:23

%S 1,2,7,40,313,3090,36767,511648,8149601,146181170,2915473799,

%T 63994242408,1532946343993,39792610701410,1112660153295487,

%U 33340011988163200,1065767723467926913,36202762585921351842,1302233685369700739399,49448677281462706745320

%N a(1) = 1, a(2) = 2; a(n+1) = 2n*a(n) - a(n-1). Symmetrically, a(n) = (a(n-1) + a(n+1))/((n-1) + (n+1)).

%H Seiichi Manyama, <a href="/A093985/b093985.txt">Table of n, a(n) for n = 1..404</a>

%F a(n) = Sum_{k = 0..floor((n-1)/2)} (-1)^k*2^(n-2*k-1)*(n-2*k-1)!*(binomial(n-k-1,k))^2. Cf. A058798. - _Peter Bala_, Aug 01 2013

%F a(n) = (Pi/2)*(Y[0, 1] * J[n, 1] - J[0, 1] * Y[n, 1]) where Y and J are Bessel functions. - _Peter Luschny_, Jan 30 2024

%e a(3)=7 because 2*2*a(2) - a(1) = 7.

%p a[1]:=1: a[2]:=2: for n from 2 to 21 do a[n+1]:=2*n*a[n]-a[n-1] od: seq(a[n],n=1..21); # _Emeric Deutsch_, Jul 31 2005

%t nxt[{n_,a_,b_}]:={n+1,b,2*n*b-a}; NestList[nxt,{2,1,2},20][[All,2]] (* _Harvey P. Dale_, Jan 09 2021 *)

%t a[n_] := (Pi/2)*(BesselY[0, 1]*BesselJ[n, 1.] - BesselJ[0, 1]*BesselY[n, 1.]);

%t Table[Round[a[n]], {n, 1, 20}] (* _Hugo Pfoertner_, Feb 12 2024 *)

%o (Magma) I:=[1,2]; [n le 2 select I[n] else 2*(n-1)*Self(n-1)-Self(n-2): n in [1..30]]; // _Vincenzo Librandi_, Aug 15 2017

%Y Cf. A058798, A093986.

%K nonn,easy

%O 1,2

%A _Amarnath Murthy_, May 22 2004

%E Corrected and extended by _Emeric Deutsch_, Jul 31 2005