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Search: id:A097681
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| A097681 |
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E.g.f.: (1/(1-x^6))*exp( 6*sum_{i>=0} x^(6*i+1)/(6*i+1) ) for an order-6 linear recurrence with varying coefficients. |
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6
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| 1, 6, 36, 216, 1296, 7776, 47376, 314496, 2612736, 28740096, 368395776, 4796983296, 60300205056, 750367328256, 10151357239296, 164475953381376, 3110937349718016, 61410199093641216, 1174438559356747776
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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) = 6*c = 0.1140186893... where c = 6*exp(psi(1/6)+EulerGamma) = 0.0190031148...(A097671), 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>=6: a(n) = 6*a(n-1) + n!/(n-6)!*a(n-6); for n<6: a(n)=6^n. E.g.f.: 1/(1-x^6)*(1+x)/(1-x)*sqrt((1+x+x^2)/(1-x+x^2))* exp(sqrt(3)*atan(sqrt(3)*x/(1-x^2))).
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EXAMPLE
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The sequence {1, 6, 36/2!, 216/3!, 1296/4!, 7776/5!, 47376/6!,...} 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^6)*exp(6*sum(i=0, n, x^(6*i+1)/(6*i+1)))+x*O(x^n), n)}
(PARI) a(n)=if(n<0, 0, if(n==0, 1, 6*a(n-1)+if(n<6, 0, n!/(n-6)!*a(n-6))))
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CROSSREFS
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Cf. A097671, A097677-A097680, A097682-A097682.
Sequence in context: A126634 A007275 A000400 this_sequence A050736 A033142 A082309
Adjacent sequences: A097678 A097679 A097680 this_sequence A097682 A097683 A097684
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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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