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%I #17 Feb 27 2021 13:20:38
%S 1,5,25,125,625,3245,19825,162125,1650625,17703125,186644425,
%T 2032320125,25569960625,382772328125,6166860390625,98093486946125,
%U 1555728351450625,26765871718953125,527380555479765625,11241893092061328125
%N E.g.f.: (1/(1-x^5))*exp( 5*sum_{i>=0} x^(5*i+1)/(5*i+1) ) for an order-5 linear recurrence with varying coefficients.
%C Limit_{n->inf} n*n!/a(n) = 5*c = 0.2247091438... where c = 5*exp(psi(1/5)+EulerGamma) = 0.0449418287...(A097667) 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>=5: a(n) = 5*a(n-1) + n!/(n-5)!*a(n-5); for n<5: a(n)=5^n. E.g.f.: B(x)*exp(C(x)) where B(x) = 1/(1-x^5)/(1-x)*(1+phi*x+x^2)^(phi/2)/(1-x/phi+x^2)^(1/phi/2) and C(x) = 5^(1/4)*sqrt(phi)*atan(5^(1/4)*sqrt(phi)*x/(2-x/phi)) + 5^(1/4)/sqrt(phi)*atan(5^(1/4)/sqrt(phi)*x/(2+phi*x)) and where phi=(sqrt(5)+1)/2.
%e The sequence {1, 5, 25/2!, 125/3!, 625/4!, 3245/5!, 19825/6!, 162125/7!,...} 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^5)*exp(5*sum(i=0,n,x^(5*i+1)/(5*i+1)))+x*O(x^n),n)}
%o (PARI) a(n)=if(n<0,0,if(n==0,1,5*a(n-1)+if(n<5,0,n!/(n-5)!*a(n-5))))
%Y Cf. A097667, A097677-A097679, A097681-A097682.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Sep 01 2004
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