A097678 - OEIS

%I #25 Feb 27 2021 13:20:38

%S 1,0,3,6,27,252,1125,10206,108297,811944,10272339,131572350,

%T 1410753267,22363938324,342373389813,4790641828518,90549635310225,

%U 1626834238205904,28073013793245603,614304628556766966,12727707975543382731

%N 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.

%C 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).

%D 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.

%D 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.

%H Vincenzo Librandi, <a href="/A097678/b097678.txt">Table of n, a(n) for n = 0..200</a>

%H Benoit Cloitre, <a href="/A097679/a097679.pdf">On a generalization of Euler-Gauss formula for the Gamma function</a>, preprint 2004.

%H Andrew Odlyzko, <a href="http://www.dtc.umn.edu/~odlyzko/doc/asymptotic.enum.pdf">Asymptotic enumeration methods</a>, in Handbook of Combinatorics, vol. 2, 1995, pp. 1063-1229.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DigammaFunction.html">Digamma Function</a>.

%F 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))).

%e 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).

%t 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 *)

%o The following PARI code generates this sequence and demonstrates

%o the general recursion with the asymptotic limit and e.g.f.:

%o /* Define Cloitre's recursion: */

%o z=[0,1,0]; r=3; s=3; zt=sum(i=1,r,z[i])

%o {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))))}

%o /* The following tends to a limit (slowly): */

%o for(n=r,20,print(if(w(n)==0,0,n^zt/w(n))*1.0,","))

%o /* This is the exact value of the limit: */

%o {s^(zt+1)*gamma(zt+1)*exp(sum(k=1,r,z[k]*(psi(k/s)+Euler)))}

%o /* Print terms w(n) multiplied by (n-r)! for e.g.f. */

%o for(n=r,20,print1((n-r)!*w(n),","))

%o /* Compare to terms generated by e.g.f.: */

%o {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))))}

%o for(n=0,20-r,print1(n!*polcoeff(EGF(x)+x*O(x^n),n),","))

%o (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)}

%o (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))))

%Y Cf. A097664, A097677, A097679-A097682.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Sep 01 2004