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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.
20
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
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
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 01 2004
STATUS
approved