A106174 a(n) = 2*n*a(n-1) - a(n-2), with a(0)=0, a(1)=1.

0, 1, 4, 23, 180, 1777, 21144, 294239, 4686680, 84066001, 1676633340, 36801867479, 881568186156, 22883970972577, 639869619046000, 19173204600407423, 612902677593991536, 20819517833595304801, 748889739331836981300

Offset: 0

Author

Roger L. Bagula, Mar 14 2006

Keywords

• nonn
• easy

Comments

Bessel recurrence at x=1: J(x,n) = (2*n/x)*J(x,n-1) - J(x,n-2).

References

• Abramowitz and Stegun, Handbook of Mathematical Functions, 9th printing, 1972, page 385.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..403
• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
• S. Janson, A divergent generating function that can be summed and analysed analytically, Discrete Mathematics and Theoretical Computer Science; 2010, Vol. 12, No. 2, 1-22.

Crossrefs

Cf. A058798.

Programs

GAP

a:=[0,1];; for n in [3..20] do a[n]:=2*(n-1)*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Mar 25 2019

Magma

I:=[0,1]; [n le 2 select I[n] else 2*(n-1)*Self(n-1) - Self(n-2): n in [1..20]]; // G. C. Greubel, Mar 25 2019

Mathematica

F[0]=0; F[1]=1; F[n_]:= F[n]= 2*n*F[n-1]-F[n-2]; Table[F[n], {n, 0, 20}]
RecurrenceTable[{a[0]==0,a[1]==1,a[n]==2n a[n-1]-a[n-2]},a,{n,20}] (* Harvey P. Dale, Oct 17 2016 *)

PARI

m=20; v=concat([0,1], vector(m-2)); for(n=3, m, v[n]=2*(n-1)*v[n-1]-v[n-2]); v \\ G. C. Greubel, Mar 25 2019

Sage

def A058798(n):
    if n < 2: return n
    return factorial(n)*2^(n-1)*hypergeometric([1/2-n/2, 1-n/2], [2, 1-n, -n], -1)
[round(A058798(n).n(164)) for n in (0..18)] # Peter Luschny, Sep 10 2014

Formula

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)*binomial(n-k,k+1), cf. A058798. - Peter Bala, Aug 01 2013

a(n) = n!*2^(n-1)*hypergeometric2F3([(1-n)/2, 1-n/2],[2, 1-n, -n], -1) for n >= 2. - Peter Luschny, Sep 10 2014

From Vaclav Kotesovec, Jun 10 2019: (Start)

a(n) = Pi*(BesselJ(1 + n, 1)*BesselY(1, 1) - BesselJ(1, 1)*BesselY(1 + n, 1))/2.

a(n) ~ BesselJ(1,1) * 2^n * n!. (End)