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A097679
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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.
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20
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1, 4, 16, 64, 280, 1600, 12160, 102400, 880000, 8358400, 94720000, 1189888000, 15213952000, 204285952000, 3092697088000, 51351519232000, 869951500288000, 15148619579392000, 287722152460288000, 5927812334878720000
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OFFSET
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0,2
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COMMENTS
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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).
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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>=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)).
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EXAMPLE
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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).
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MATHEMATICA
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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 *)
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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=[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
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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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