A058308 a(n) = (n+2)*a(n-1) + a(n-2), with a(0)=0, a(1)=1.

0, 1, 4, 21, 130, 931, 7578, 69133, 698908, 7757121, 93784360, 1226953801, 17271137574, 260294017411, 4181975416150, 71353876091961, 1288551745071448, 24553837032449473, 492365292394060908, 10364224977307728541, 228505314793164088810

Offset: 0

Author

N. J. A. Sloane, Dec 09 2000

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..445
• 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.
• Russell Walsmith, CL-Chemy II: Reflections and Other Symmetries

Crossrefs

A column of A058294.

Similar recurrences: A001040, A001053, A058279, A058307, A093858. - Wolfdieter Lang, May 19 2010

Cf. A058798.

Programs

Magma

I:=[1,4]; [0] cat [n le 2 select I[n] else (n+2)*Self(n-1) +Self(n-2): n in [1..30]]; // G. C. Greubel, Nov 24 2018

Mathematica

RecurrenceTable[{a[0]==0,a[1]==1,a[n]==a[n-2]+(n+2)a[n-1]},a[n], {n,20}] (* Harvey P. Dale, May 21 2011 *)
FullSimplify[Table[(-4*BesselI[3+n,-2]*BesselK[4,2] + BesselI[3+n,-2]*BesselK[5,2] + 4*BesselI[4,2]*BesselK[3+n,2] + BesselI[5,2]*BesselK[3+n,2]) / (BesselI[5,2]*BesselK[4,2] + BesselI[4,2]*BesselK[5,2]),{n,0,20}]] (* Vaclav Kotesovec, Oct 05 2013 *)

PARI

m=30; v=concat([1,4], vector(m-2)); for(n=3, m, v[n]=(n+2)*v[n-1] +v[n-2]); concat([0], v) \\ G. C. Greubel, Nov 24 2018

Sage

def A058308(n):
    if n < 2: return n
    return factorial(n+2)*hypergeometric([1/2-n/2, 1-n/2], [4, -n-2, 1-n], 4)/6
[round(A058308(n).n(100)) for n in (0..20)] # Peter Luschny, Sep 10 2014

Sage

@cached_function
def A058308(n):
    if n==0: return 0
    if n==1: return 1
    return (n+2)*A058308(n-1) + A058308(n-2)
[A058308(n) for n in range(30)]  # G. C. Greubel, Nov 24 2018

Formula

E.g.f.: -Pi*(BesselY(4, 2*i)*BesselI(3, 2*sqrt(1-x)) - i*BesselI(4, 2)*BesselY(3, 2*i*sqrt(1-x)))/(1-x)^(3/2). Such e.g.f. computations were the result of an e-mail exchange with Gary Detlefs. After differentiation and setting x=0 one has to use simplifications. See the Abramowitz-Stegun handbook, p. 360, 9.1.16 and p. 375, 9.63. - Wolfdieter Lang, May 19 2010

a(n) = Sum_{k = 0..floor((n-1)/2)} (n-2*k-1)!*binomial(n-k-1,k)* binomial(n-k+2,k+3). Cf. A058798. - Peter Bala, Aug 01 2013

a(n) = (-4*BesselI(3+n,-2)*BesselK(4,2) + BesselI(3+n,-2)*BesselK(5,2) + 4*BesselI(4,2)*BesselK(3+n,2) + BesselI(5,2)*BesselK(3+n,2)) / (BesselI(5,2)*BesselK(4,2) + BesselI(4,2)*BesselK(5,2)). - Vaclav Kotesovec, Oct 05 2013

a(n) ~ (4*BesselI(4,2) + BesselI(5,2))/(BesselI(5,2)*BesselK(4,2) + BesselI(4,2)*BesselK(5,2)) * sqrt(Pi/2) * n^(n+5/2)/exp(n). - Vaclav Kotesovec, Oct 05 2013

a(n) = (n+2)!*hypergeometric([1/2-n/2,1-n/2],[4,-n-2,1-n],4)/6 for n >= 2. - Peter Luschny, Sep 10 2014

a(n) = (n+1)! [x^(n+1)] (2/(1-x))*(K(3,2)*I(2,2*sqrt(1-x))+I(3,2)*K(2,2*sqrt(1-x))), K and I Bessel functions. - Peter Luschny, May 01 2017