login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A058309 a(n) = (n+3)*a(n-1) + a(n-2), with a(0)=0, a(1)=1. 6
0, 1, 5, 31, 222, 1807, 16485, 166657, 1849712, 22363201, 292571325, 4118361751, 62067997590, 997206323191, 17014575491837, 307259565176257, 5854946313840720, 117406185841990657, 2471384848995644517, 54487872863746170031, 1255692460715157555230 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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, DCL-Chemy II: Reflections and Other Symmetries

FORMULA

a(n) = Sum_{k = 0..floor((n-1)/2)} (n-2*k-1)!*binomial(n-k-1,k)* binomial(n-k+3,k+4). Cf. A058798. - Peter Bala, Aug 01 2013

a(n) = (I(n+4,-2)*(5*K_5-K_6)+K(n+4,2)*(5*I_5+I_6))/(I_6*K_5+I_5* K_6), where I_n and K_n are the Bessel functions of the first respectively the second kind, all evaluated at x=2. - Peter Luschny, Sep 11 2014

a(n) = (n+3)!*hypergeometric([1/2-n/2, 1-n/2], [5, -n-3, 1-n], 4)/24 for n >= 2. - Peter Luschny, Sep 12 2014

0 = a(n)*(+a(n+2)) + a(n+1)*(-a(n+1) - a(n+2) + a(n+3)) + a(n+2)*(-a(n+2)) for all n in Z. - Michael Somos, Sep 13 2014

a(-8-n) = a(n) for all n in Z. - Michael Somos, Sep 13 2014

EXAMPLE

G.f. = x + 5*x^2 + 31*x^3 + 222*x^4 + 1807*x^5 + 16485*x^6 + 166657*x^7 + ...

MAPLE

bi:=n->BesselI(4+n, -2); bk:=n->BesselK(4+n, 2); i:=n->BesselI(n, 2);

k:=n->BesselK(n, 2); a := n ->(bi(n)*(5*k(5)-k(6))+bk(n)*(5*i(5) +i(6)))/(i(6)*k(5)+i(5)*k(6)); seq(round(evalf(a(n), 99)), n=0..20); # Peter Luschny, Sep 11 2014

MATHEMATICA

a[0] = 0; a[1] = 1; a[n_] := a[n - 2] + (n + 3)*a[n - 1]; Table[a[n], {n, 0, 20}] (* Wesley Ivan Hurt, Sep 12 2014 *)

a[ n_] := With[{m = n + 4}, 2 (BesselK[m, 2] BesselI[4, 2] - (-1)^m BesselI[m, 2] BesselK[4, 2]) // FullSimplify]; (* Michael Somos, Dec 09 2014 *)

a[ n_] := With[{m = Abs[n + 4]}, If[ m < 5, {-10, 7, -3, 1, 0}[[m + 1]], (m - 1)! HypergeometricPFQ[ {5/2 - m/2, 3 - m/2}, {5, 1 - m, 5 - m}, 4] / 24]]; (* Michael Somos, Dec 09 2014 *)

PROG

(PARI) a(n)=round((besseli(n+4, -2)*(5*besselk(5, 2)-besselk(6, 2)) + besselk(n+4, 2)*(5*besseli(5, 2) + besseli(6, 2))) / (besseli(6, 2)*besselk(5, 2) + besseli(5, 2)*besselk(6, 2))) \\ Charles R Greathouse IV, Sep 11 2014

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

(Sage)

def A058309(n):

    if n < 2: return n

    return factorial(n+3)*hypergeometric([1/2-n/2, 1-n/2], [5, -n-3, 1-n], 4)/24

[round(A058309(n).n(100)) for n in (0..20)] # Peter Luschny, Sep 12 2014

(MAGMA) I:=[1, 5]; [0] cat [n le 2 select I[n] else (n+3)*Self(n-1) +Self(n-2): n in [1..30]]; // G. C. Greubel, Nov 24 2018

(Sage)

def A058309(n, D={}):

    if D.has_key(n):

        return D[n]

    else:

        if (n==0): result = 0

        elif (n==1): result = 1

        else: result = expand((n+3)*A058309(n-1) + A058309(n-2))

    D[n] = result

    return result

[A058309(n) for n in range(30)] # G. C. Greubel, Nov 24 2018

CROSSREFS

A column of A058294. Cf. A058798.

Sequence in context: A143020 A059035 A199877 * A226924 A192950 A001910

Adjacent sequences:  A058306 A058307 A058308 * A058310 A058311 A058312

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Dec 09 2000

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 9 19:51 EST 2019. Contains 329879 sequences. (Running on oeis4.)