A097678 E.g.f.: (1/(1-x^3))*exp( 3*sum_{i>=0} x^(3*i+2)/(3*i+2) ) for an order-3 linear recurrence with varying coefficients.

1, 0, 3, 6, 27, 252, 1125, 10206, 108297, 811944, 10272339, 131572350, 1410753267, 22363938324, 342373389813, 4790641828518, 90549635310225, 1626834238205904, 28073013793245603, 614304628556766966, 12727707975543382731

Offset: 0

Author

Paul D. Hanna, Sep 01 2004

Keywords

• nonn

Comments

Limit_{n->inf} n*n!/a(n) = 3*c = 4.2896529252... where c = 3*exp(psi(2/3)+EulerGamma) = 1.4298843084...(A097664) and EulerGamma is the Euler-Mascheroni constant (A001620) and psi() is the Digamma function (see Mathworld link).

Examples

			The sequence {1, 0, 3/2!, 6/3!, 27/4!, 252/5!, 1125/6!, 10206/7!,...} 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

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• 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. A097664, A097677, A097679-A097682.

Programs

Mathematica

CoefficientList[Series[1/Sqrt[(1-x^3)*(1-x)^3]*E^(-Sqrt[3] * ArcTan[Sqrt[3] * x/(2+x)]), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Feb 15 2014 *)

PARI

{a(n)=n!*polcoeff(1/(1-x^3)*exp(3*sum(i=0,n,x^(3*i+2)/(3*i+2)))+x*O(x^n),n)}

PARI

a(n)=if(n<0,0,if(n==0,1,3*(n-1)*a(n-2)+if(n<3,0,n!/(n-3)!*a(n-3))))

Formula

For n>=3: a(n) = 3*(n-1)*a(n-2) + n!/(n-3)!*a(n-3); a(0)=1, a(1)=0, a(2)=3. E.g.f.: 1/sqrt((1-x^3)*(1-x)^3)*exp(-sqrt(3)*atan(sqrt(3)*x/(2+x))).