0,3

Limit_{n->inf} n*n!/a(n) = 3*c = 4.2896529252... where c = 3*exp(psi(2/3)+EulerGamma) = 1.4298843084...(A097664) and EulerGamma is the Euler-Mascheroni constant (A001620) and psi() is the Digamma function (see Mathworld link).

A. M. Odlyzko, Linear recurrences with varying coefficients, in Handbook of Combinatorics, Vol. 2, R. L. Graham, M. Grotschel and L. Lovasz, eds., Elsevier, Amsterdam, 1995, pp. 1135-1138.

Benoit Cloitre, On a generalization of Euler-Gauss formula for the Gamma function, pre-print 2004.

Andrew Odlyzko, Asymptotic enumeration methods, in Handbook of Combinatorics, vol. 2, 1995, pp. 1063-1229.

Eric Weisstein's World of Mathematics, Digamma Function.

For n>=3: a(n) = 3*(n-1)*a(n-2) + n!/(n-3)!*a(n-3); a(0)=1, a(1)=0, a(2)=3. E.g.f.: 1/sqrt((1-x^3)*(1-x)^3)*exp(-sqrt(3)*atan(sqrt(3)*x/(2+x))).

The sequence {1, 0, 3/2!, 6/3!, 27/4!, 252/5!, 1125/6!, 10206/7!,...}

is generated by a recursion described by Benoit Cloitre's generalized

Euler-Gauss formula for the Gamma function (see Cloitre link).

The following PARI code generates this sequence and demonstrates

the general recursion with the asymptotic limit and e.g.f.:

/* Define Cloitre's recursion: */

z=[0, 1, 0]; r=3; s=3; zt=sum(i=1, r, z[i])

{w(n)=if(n<r, 0, if(n==r, 1, w(n-s)+s/(n-r)*sum(i=1, r, z[i]*w(n-i))))}

/* The following tends to a limit (slowly): */

for(n=r, 20, print(if(w(n)==0, 0, n^zt/w(n))*1.0, ", "))

/* This is the exact value of the limit: */

{s^(zt+1)*gamma(zt+1)*exp(sum(k=1, r, z[k]*(psi(k/s)+Euler)))}

/* Print terms w(n) multiplied by (n-r)! for e.g.f. */

for(n=r, 20, print1((n-r)!*w(n), ", "))

/* Compare to terms generated by e.g.f.: */

{EGF(x)=1/(1-x^s)*exp(s*sum(i=0, 30, sum(j=1, r, z[j]*x^(s*i+j)/(s*i+j))))}

for(n=0, 20-r, print1(n!*polcoeff(EGF(x)+x*O(x^n), n), ", "))

(PARI) {a(n)=n!*polcoeff(1/(1-x^3)*exp(3*sum(i=0, n, x^(3*i+2)/(3*i+2)))+x*O(x^n), n)}

(PARI) a(n)=if(n<0, 0, if(n==0, 1, 3*(n-1)*a(n-2)+if(n<3, 0, n!/(n-3)!*a(n-3))))

Cf. A097664, A097677, A097679-A097682.

Sequence in context: A023169 A083695 A060170 this_sequence A074894 A083675 A085076

Adjacent sequences: A097675 A097676 A097677 this_sequence A097679 A097680 A097681

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Sep 01 2004