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A097678
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E.g.f.: (1/(1-x^3))*exp( 3*sum_{i>=0} x^(3*i+2)/(3*i+2) ) for an order-3 linear recurrence with varying coefficients.
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4
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1, 0, 3, 6, 27, 252, 1125, 10206, 108297, 811944, 10272339, 131572350, 1410753267, 22363938324, 342373389813, 4790641828518, 90549635310225, 1626834238205904, 28073013793245603, 614304628556766966, 12727707975543382731
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OFFSET
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0,3
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COMMENTS
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Limit_{n->inf} n*n!/a(n) = 3*c = 4.2896529252... where c = 3*exp(psi(2/3)+EulerGamma) = 1.4298843084...(A097664) and EulerGamma is the Euler-Mascheroni constant (A001620) and psi() is the Digamma function (see Mathworld link).
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REFERENCES
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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.
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LINKS
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FORMULA
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For n>=3: a(n) = 3*(n-1)*a(n-2) + n!/(n-3)!*a(n-3); a(0)=1, a(1)=0, a(2)=3. E.g.f.: 1/sqrt((1-x^3)*(1-x)^3)*exp(-sqrt(3)*atan(sqrt(3)*x/(2+x))).
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EXAMPLE
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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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MATHEMATICA
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CoefficientList[Series[1/Sqrt[(1-x^3)*(1-x)^3]*E^(-Sqrt[3] * ArcTan[Sqrt[3] * x/(2+x)]), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Feb 15 2014 *)
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PROG
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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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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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