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

0,3

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).

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.

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))).

Example

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).

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 *)

Prog

The following PARI code generates this sequence and demonstrates
the general recursion with the asymptotic limit and e.g.f.:
/* Define Cloitre's recursion: */
z=[0, 1, 0]; r=3; s=3; zt=sum(i=1, r, z[i])
{w(n)=if(n<r, 0, if(n==r, 1, w(n-s)+s/(n-r)*sum(i=1, r, z[i]*w(n-i))))}
/* The following tends to a limit (slowly): */
for(n=r, 20, print(if(w(n)==0, 0, n^zt/w(n))*1.0, ", "))
/* This is the exact value of the limit: */
{s^(zt+1)*gamma(zt+1)*exp(sum(k=1, r, z[k]*(psi(k/s)+Euler)))}
/* Print terms w(n) multiplied by (n-r)! for e.g.f. */
for(n=r, 20, print1((n-r)!*w(n), ", "))
/* Compare to terms generated by e.g.f.: */
{EGF(x)=1/(1-x^s)*exp(s*sum(i=0, 30, sum(j=1, r, z[j]*x^(s*i+j)/(s*i+j))))}
for(n=0, 20-r, print1(n!*polcoeff(EGF(x)+x*O(x^n), n), ", "))
(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))))

Crossrefs

Cf. A097664, A097677, A097679-A097682.

Sequence in context: A223143 A372039 A270889 * A251609 A366560 A074894

Adjacent sequences: A097675 A097676 A097677 * A097679 A097680 A097681

Keyword

nonn

Author

Paul D. Hanna, Sep 01 2004

Status

approved