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 A097680 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. 6
 1, 5, 25, 125, 625, 3245, 19825, 162125, 1650625, 17703125, 186644425, 2032320125, 25569960625, 382772328125, 6166860390625, 98093486946125, 1555728351450625, 26765871718953125, 527380555479765625, 11241893092061328125 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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). REFERENCES 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. LINKS Benoit Cloitre, On a generalization of Euler-Gauss formula for the Gamma function, preprint 2004. Andrew Odlyzko, Asymptotic enumeration methods, in Handbook of Combinatorics, vol. 2, 1995, pp. 1063-1229. Eric Weisstein's World of Mathematics, Digamma Function. FORMULA 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. EXAMPLE 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). PROG (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)} (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)))) CROSSREFS Cf. A097667, A097677-A097679, A097681-A097682. Sequence in context: A050735 A195948 A083590 * A069030 A111993 A341266 Adjacent sequences:  A097677 A097678 A097679 * A097681 A097682 A097683 KEYWORD nonn AUTHOR Paul D. Hanna, Sep 01 2004 STATUS approved

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Last modified May 29 07:49 EDT 2022. Contains 354122 sequences. (Running on oeis4.)