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)).

1, 1, 3, 17, 133, 1313, 15623, 217409, 3462921, 62115169, 1238840459, 27192374929, 651378157837, 16908639728833, 472790534249487, 14166807387755777, 452865045873935377, 15383244752326047041, 553343946037863758099, 21011686704686496760721

Offset: 1

Author

Amarnath Murthy, May 22 2004

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..200
• The Wolfram Functions Site, BesselJ[nu,z], Recurrence identities

Crossrefs

Cf. A093985.

Programs

Maple

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
# second Maple program:
a:= proc(n) a(n):= `if`(n<2, 1, a(n-1)*(2*n-2)-a(n-2)) end:
seq(a(n), n=1..25);  # Alois P. Heinz, May 21 2013

Mathematica

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}]
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 *)
RecurrenceTable[{a[1]==a[2]==1,a[n+1]==2n a[n]-a[n-1]},a,{n,20}] (* Harvey P. Dale, Jan 17 2021 *)

Formula

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

a(n) ~ sqrt(Pi/2)*(BesselJ(0,1)-BesselJ(1,1)) * 2^n*n^(n-1/2)*exp(-n). - Vaclav Kotesovec, Aug 13 2013

a(-n) = (-1)^n * a(n). - Michael Somos, May 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)). - Michael Somos, May 10 2014

Extensions

More terms from Emeric Deutsch, Apr 17 2005