A097682 E.g.f.: (1/(1-x^8))*exp( 8*sum_{i>=0} x^(8*i+1)/(8*i+1) ) for an order-8 linear recurrence with varying coefficients.

1, 8, 64, 512, 4096, 32768, 262144, 2097152, 16817536, 137443328, 1215668224, 13131579392, 186802241536, 3194809745408, 57299125141504, 1002518381330432, 16747075923705856, 268695698674024448, 4294396462470529024

Offset: 0

Author

Paul D. Hanna, Sep 01 2004

Keywords

• nonn

Comments

Limit_{n->inf} n*n!/a(n) = 8*c = 0.0259289826... where c = 8*exp(psi(1/8)+EulerGamma) = 0.0032411228...(A097673) and EulerGamma is the Euler-Mascheroni constant (A001620) and psi() is the Digamma function (see Mathworld link).

Examples

			The sequence {1, 8, 64/2!, 512/3!, 4096/4!, 32768/5!, 262144/6!,...} is generated by a recursion described by Benoit Cloitre's generalized Euler-Gauss formula for the Gamma function (see Cloitre 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.

Crossrefs

Cf. A097673, A097677-A097681.

Programs

PARI

{a(n)=n!*polcoeff(1/(1-x^8)*exp(8*sum(i=0,n,x^(8*i+1)/(8*i+1)))+x*O(x^n),n)}

PARI

a(n)=if(n<0,0,if(n==0,1,8*a(n-1)+if(n<8,0,n!/(n-8)!*a(n-8))))

Formula

For n>=8: a(n) = 8*a(n-1) + n!/(n-8)!*a(n-8); for n<8: a(n)=8^n. E.g.f.: 1/(1-x^8)*(1+x)/(1-x)* ((1+sqrt(2)*x+x^2)/(1-sqrt(2)*x+x^2))^(1/sqrt(2))* exp(sqrt(2)*atan(sqrt(2)*x/(1-x^2))+2*atan(x)).