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Search: id:A097680
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| A097680 |
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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. |
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7
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| 1, 5, 25, 125, 625, 3245, 19825, 162125, 1650625, 17703125, 186644425, 2032320125, 25569960625, 382772328125, 6166860390625, 98093486946125, 1555728351450625, 26765871718953125, 527380555479765625, 11241893092061328125
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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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).
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REFERENCES
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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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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.
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FORMULA
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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.
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EXAMPLE
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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).
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PROGRAM
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(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))))
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CROSSREFS
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Cf. A097667, A097677-A097679, A097681-A097682.
Sequence in context: A000351 A050735 A083590 this_sequence A069030 A111993 A113996
Adjacent sequences: A097677 A097678 A097679 this_sequence A097681 A097682 A097683
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KEYWORD
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nonn
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Sep 01 2004
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