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%I #13 Feb 27 2021 13:12:00
%S 1,8,64,512,4096,32768,262144,2097152,16817536,137443328,1215668224,
%T 13131579392,186802241536,3194809745408,57299125141504,
%U 1002518381330432,16747075923705856,268695698674024448,4294396462470529024
%N E.g.f.: (1/(1-x^8))*exp( 8*sum_{i>=0} x^(8*i+1)/(8*i+1) ) for an order-8 linear recurrence with varying coefficients.
%C Limit_{n->inf} n*n!/a(n) = 8*c = 0.0259289826... where c = 8*exp(psi(1/8)+EulerGamma) = 0.0032411228...(A097673) and EulerGamma is the Euler-Mascheroni constant (A001620) and psi() is the Digamma function (see Mathworld link).
%D 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.
%D 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.
%H Benoit Cloitre, <a href="/A097679/a097679.pdf">On a generalization of Euler-Gauss formula for the Gamma function</a>, Preprint 2004.
%H Andrew Odlyzko, <a href="http://www.dtc.umn.edu/~odlyzko/doc/asymptotic.enum.pdf">Asymptotic enumeration methods</a>, in Handbook of Combinatorics, vol. 2, 1995, pp. 1063-1229.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DigammaFunction.html">Digamma Function</a>.
%F For n>=8: a(n) = 8*a(n-1) + n!/(n-8)!*a(n-8); for n<8: a(n)=8^n. E.g.f.: 1/(1-x^8)*(1+x)/(1-x)* ((1+sqrt(2)*x+x^2)/(1-sqrt(2)*x+x^2))^(1/sqrt(2))* exp(sqrt(2)*atan(sqrt(2)*x/(1-x^2))+2*atan(x)).
%e The sequence {1, 8, 64/2!, 512/3!, 4096/4!, 32768/5!, 262144/6!,...} is generated by a recursion described by Benoit Cloitre's generalized Euler-Gauss formula for the Gamma function (see Cloitre link).
%o (PARI) {a(n)=n!*polcoeff(1/(1-x^8)*exp(8*sum(i=0,n,x^(8*i+1)/(8*i+1)))+x*O(x^n),n)}
%o (PARI) a(n)=if(n<0,0,if(n==0,1,8*a(n-1)+if(n<8,0,n!/(n-8)!*a(n-8))))
%Y Cf. A097673, A097677-A097681.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Sep 01 2004
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