A097677 E.g.f.: (1/(1-x^3))*exp( 3*sum_{i>=0} x^(3*i+1)/(3*i+1) ) for an order-3 linear recurrence with varying coefficients.

1, 3, 9, 33, 171, 1053, 7119, 57267, 525609, 5164803, 56726649, 690532857, 8889138531, 124010345277, 1880154795519, 29907812576187, 506398197859281, 9190226159295363, 173999328850897641, 3466197108906552657

Offset: 0

Author

Paul D. Hanna, Sep 01 2004

Keywords

• nonn

Comments

Limit_{n->inf} n*n!/a(n) = 3*c = 0.6993572795... where c = 3*exp(psi(1/3)+EulerGamma) = 0.2331190931...(A097663) and EulerGamma is the Euler-Mascheroni constant (A001620) and psi() is the Digamma function.

Examples

			The sequence {1, 3, 9/2!, 33/3!, 171/4!, 1053/5!, 7119/6!, 57267/7!,...} 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. A097663, A097678-A097682.

Programs

PARI

{a(n)=n!*polcoeff(1/(1-x^3)*exp(3*sum(i=0,n,x^(3*i+1)/(3*i+1)))+x*O(x^n),n)}

PARI

a(n)=if(n<0,0,if(n==0,1,3*a(n-1)+if(n<3,0,n!/(n-3)!*a(n-3))))

Formula

For n>=3: a(n) = 3*a(n-1) + n!/(n-3)!*a(n-3); for n<3: a(n)=3^n. E.g.f.: 1/sqrt((1-x^3)*(1-x)^3)*exp(sqrt(3)*atan(sqrt(3)*x/(2+x))).