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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

Table of n, a(n) for n=0..19.

Benoit Cloitre, On a generalization of Euler-Gauss formula for the Gamma function, pre-print 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 A113996

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 November 21 12:28 EST 2019. Contains 329370 sequences. (Running on oeis4.)