0,2

Limit_{n->inf} n*n!/a(n) = 4*c = 0.4157591527... where c = 4*exp(psi(1/4)+EulerGamma) = 0.1039397881...(A097665), 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>=4: a(n) = 4*a(n-1) + n!/(n-4)!*a(n-4); for n<4: a(n)=4^n. E.g.f.: (1+x)/(1-x^4)/(1-x)*exp(2*atan(x)).

The sequence {1, 4, 16/2!, 64/3!, 280/4!, 1600/5!, 12160/6!, 102400/7!,...} is generated by a recursion described by Benoit Cloitre's generalized Euler-Gauss formula for the Gamma function (see Cloitre link).

Range[0, 20]! CoefficientList[ Series[ E^(4Sum[x^(4k + 1)/(4k + 1), {k, 0, 150}])/(1 - x^4), {x, 0, 20}], x] (from Robert G. Wilson v Sep 03 2004)

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, 0]; r=4; s=4; 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), ", "))

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

(PARI) {a(n)=n!*polcoeff(1/(1-x^4)*exp(4*sum(i=0, n, x^(4*i+1)/(4*i+1)))+x*O(x^n), n)}

(PARI) a(n)=if(n<0, 0, if(n==0, 1, 4*a(n-1)+if(n<4, 0, n!/(n-4)!*a(n-4))))

Cf. A097665, A097677, A097678, A097680, A097681, A097682.

Adjacent sequences: A097676 A097677 A097678 this_sequence A097680 A097681 A097682

Sequence in context: A098590 A071357 A113995 this_sequence A005401 A002923 A013149

nonn

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