0,2

Limit_{n->inf} n*n!/a(n) = 3*c = 0.6993572795... where c = 3*exp(psi(1/3)+EulerGamma) = 0.2331190931...(A097663), and EulerGamma is the Euler-Mascheroni constant (A001620), and psi() is the Digamma function.

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

The sequence {1, 3, 9/2!, 33/3!, 171/4!, 1053/5!, 7119/6!, 57267/7!,...}

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=[1, 0, 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(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), ", "))

/* -----------------------END---------------------- */

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

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

Cf. A097663, A097678-A097682.

Adjacent sequences: A097674 A097675 A097676 this_sequence A097678 A097679 A097680

Sequence in context: A101899 A009220 A007489 this_sequence A138769 A100076 A053791

nonn

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