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A093986 a(1) = 1, a(2) = 1, a(n+1) = 2n*a(n) - a(n-1). Symmetrically a(n) = (a(n-1) + a(n+1))/((n-1) + (n+1)).

%I A093986 #28 Jan 17 2021 18:27:40
%S A093986 1,1,3,17,133,1313,15623,217409,3462921,62115169,1238840459,
%T A093986 27192374929,651378157837,16908639728833,472790534249487,
%U A093986 14166807387755777,452865045873935377,15383244752326047041,553343946037863758099,21011686704686496760721
%N A093986 a(1) = 1, a(2) = 1, a(n+1) = 2n*a(n) - a(n-1). Symmetrically a(n) = (a(n-1) + a(n+1))/((n-1) + (n+1)).
%H A093986 Alois P. Heinz, <a href="/A093986/b093986.txt">Table of n, a(n) for n = 1..200</a>
%H A093986 <a href="http://functions.wolfram.com/Bessel-TypeFunctions/BesselJ/17/01/01/">The Wolfram Functions Site, BesselJ[nu,z], Recurrence identities</a>
%F A093986 a(n) = ((Y_0(1) - Y_1(1))*J_n(1) + (J_1(1)-J_0(1))*Y_n(1))*Pi/2, where J_n(x) and Y_n(x) are the Bessel function of the first and second kind. - _Vladimir Reshetnikov_, May 21 2013
%F A093986 a(n) ~ sqrt(Pi/2)*(BesselJ(0,1)-BesselJ(1,1)) * 2^n*n^(n-1/2)*exp(-n). - _Vaclav Kotesovec_, Aug 13 2013
%F A093986 a(-n) = (-1)^n * a(n). - _Michael Somos_, May 10 2014
%F A093986 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)). - _Michael Somos_, May 10 2014
%p A093986 a[1]:=1:a[2]:=1:for n from 3 to 21 do a[n]:=2*(n-1)*a[n-1]-a[n-2] od: seq(a[n],n=1..21); # _Emeric Deutsch_, Apr 17 2005
%p A093986 # second Maple program:
%p A093986 a:= proc(n) a(n):= `if`(n<2, 1, a(n-1)*(2*n-2)-a(n-2)) end:
%p A093986 seq(a(n), n=1..25);  # _Alois P. Heinz_, May 21 2013
%t A093986 Table[DifferenceRoot[Function[{a, n}, {a[n] - 2*(n + 1)*a[n + 1] + a[n + 2] == 0, a[0] == 1, a[1] == 1}]][n], {n, 1, 20}]
%t A093986 Table[FullSimplify[((BesselY[0, 1] - BesselY[1, 1]) BesselJ[n, 1] + (BesselJ[1, 1] - BesselJ[0, 1]) BesselY[n, 1]) Pi/2], {n, 1, 20}] (* _Vladimir Reshetnikov_, May 21 2013 *)
%t A093986 RecurrenceTable[{a[1]==a[2]==1,a[n+1]==2n a[n]-a[n-1]},a,{n,20}] (* _Harvey P. Dale_, Jan 17 2021 *)
%Y A093986 Cf. A093985.
%K A093986 nonn
%O A093986 1,3
%A A093986 _Amarnath Murthy_, May 22 2004
%E A093986 More terms from _Emeric Deutsch_, Apr 17 2005