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

0,2

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.

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

Table of n, a(n) for n=0..19.

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*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))).

Example

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).

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=[1, 0, 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(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), ", "))
/* -----------------------END---------------------- */
(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))))

Crossrefs

Cf. A097663, A097678-A097682.

Sequence in context: A294638 A201968 A264237 * A138769 A100076 A213907

Adjacent sequences: A097674 A097675 A097676 * A097678 A097679 A097680

Keyword

nonn

Author

Paul D. Hanna, Sep 01 2004

Status

approved