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REFERENCES
| 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.
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EXAMPLE
| The sequence {1, 0, 3/2!, 6/3!, 27/4!, 252/5!, 1125/6!, 10206/7!,...}
is generated by a recursion described by Benoit Cloitre's generalized
Euler-Gauss formula for the Gamma function (see Cloitre link).
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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=[0, 1, 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(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), ", "))
(PARI) {a(n)=n!*polcoeff(1/(1-x^3)*exp(3*sum(i=0, n, x^(3*i+2)/(3*i+2)))+x*O(x^n), n)}
(PARI) a(n)=if(n<0, 0, if(n==0, 1, 3*(n-1)*a(n-2)+if(n<3, 0, n!/(n-3)!*a(n-3))))
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