A097679 E.g.f.: (1/(1-x^4))*exp( 4*Sum_{i>=0} x^(4*i+1)/(4*i+1) ) for an order-4 linear recurrence with varying coefficients.

1, 4, 16, 64, 280, 1600, 12160, 102400, 880000, 8358400, 94720000, 1189888000, 15213952000, 204285952000, 3092697088000, 51351519232000, 869951500288000, 15148619579392000, 287722152460288000, 5927812334878720000

Offset

0,2

Comments

Lim_{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).

References

Mohammad K. Azarian, Problem 1218, Pi Mu Epsilon Journal, Vol. 13, No. 2, Spring 2010, p. 116. Solution published in Vol. 13, No. 3, Fall 2010, pp. 183-185.

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.

Links

Table of n, a(n) for n=0..19.

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

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

Eric Weisstein's World of Mathematics, Digamma Function.

Formula

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)).

Example

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).

Mathematica

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

Prog

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))))
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!((1+x)/(1-x^4)/(1-x)*Exp(2*Arctan(x)))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 29 2018

Crossrefs

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

Sequence in context: A113995 A212443 A294036 * A158778 A227313 A155518

Adjacent sequences: A097676 A097677 A097678 * A097680 A097681 A097682

Keyword

nonn

Author

Paul D. Hanna, Sep 01 2004

Status

approved