OFFSET
0,2
COMMENTS
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
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>=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))).
EXAMPLE
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).
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]; 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))))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 01 2004
STATUS
approved