A121353 a(n) = (3*n - 2)*a(n-1) - a(n-2) starting a(0)=0, a(1)=1.

0, 1, 4, 27, 266, 3431, 54630, 1034539, 22705228, 566596161, 15841987280, 490535009519, 16662348336366, 616016353436023, 24623991789104554, 1058215630578059799, 48653295014801646200, 2382953240094702604001, 123864915189909733761852, 6810187382204940654297859

Offset: 0

Author

Roger L. Bagula and Bob Hanlon (hanlonr(AT)cox.net), Sep 05 2006

Keywords

• nonn
• easy

Comments

In the hypergeometric family a(n) = (a0*n+c0)*a(n-1)+b0*a(n-2) we have A053984, A058797, A121323, A121351, and this here with a0=3, where a(n) can be expressed in a characteristic cross-product of Bessel functions.

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..381
• 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. A053984, A058798, A121351, A121323.

Programs

Mathematica

f[n_Integer] = Module[{a}, a[n] /. RSolve[{a[n] == (3*n - 2)*a[n - 1] - a[n - 2], a[0] == 0, a[1] == 1}, a[n], n][[1]] // FullSimplify] Rationalize[N[Table[f[n], {n, 0, 25}], 100], 0]
RecurrenceTable[{a[0]==0, a[1]==1, a[n]==(3n-2)*a[n-1]-a[n-2]}, a, {n, 20}]  (* Vaclav Kotesovec, Jul 31 2014 *)
nxt[{n_,a_,b_}]:={n+1,b,b(3n+1)-a}; NestList[nxt,{1,0,1},20][[;;,2]] (* Harvey P. Dale, Jun 03 2023 *)

Sage

def A121353(n):
    if n < 2: return n
    return 3^n*gamma(n+1/3)*hypergeometric([1/2-n/2,1-n/2], [4/3, 2/3 -n, 1-n], -4/9)/gamma(1/3)
[round(A121353(n).n(100)) for n in (0..19)] # Peter Luschny, Sep 10 2014

Formula

a(n) = (Pi/3) * (BesselJ(1/3+n,2/3) * BesselY(1/3,2/3) - BesselJ(1/3,2/3) * BesselY(1/3+n,2/3)).

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

a(n) ~ n! * BesselJ(1/3, 2/3) * 3^(n-2/3) * n^(-2/3). - Vaclav Kotesovec, Jul 31 2014

a(n) = 3^n*Gamma(n+1/3)*hypergeometric([1/2-n/2, 1-n/2], [4/3, 2/3-n, 1-n], -4/9)/Gamma(1/3) for n >= 2. - Peter Luschny, Sep 10 2014