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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.

%I A097677 #18 Feb 16 2025 08:32:54
%S A097677 1,3,9,33,171,1053,7119,57267,525609,5164803,56726649,690532857,
%T A097677 8889138531,124010345277,1880154795519,29907812576187,506398197859281,
%U A097677 9190226159295363,173999328850897641,3466197108906552657
%N 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.
%C A097677 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.
%D A097677 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 A097677 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 A097677 Benoit Cloitre, <a href="/A097679/a097679.pdf">On a generalization of Euler-Gauss formula for the Gamma function</a>, preprint 2004.
%H A097677 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 A097677 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DigammaFunction.html">Digamma Function</a>.
%F A097677 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))).
%e A097677 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).
%o A097677 The following PARI code generates this sequence and demonstrates
%o A097677 the general recursion with the asymptotic limit and e.g.f.:
%o A097677 /* ------------------------------------------------ */
%o A097677 /* Define Cloitre's recursion: */
%o A097677 z=[1,0,0]; r=3; s=3; zt=sum(i=1,r,z[i])
%o A097677 {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 A097677 /* ------------------------------------------------ */
%o A097677 /* The following tends to a limit (slowly): */
%o A097677 for(n=r,20,print(n^zt/w(n)*1.0,","))
%o A097677 /* This is the exact value of the limit: */
%o A097677 {s^(zt+1)*gamma(zt+1)*exp(sum(k=1,r,z[k]*(psi(k/s)+Euler)))}
%o A097677 /* ------------------------------------------------ */
%o A097677 /* Print terms w(n) multiplied by (n-r)! for e.g.f. */
%o A097677 for(n=r,20,print1((n-r)!*w(n),","))
%o A097677 /* Compare to terms generated by e.g.f.: */
%o A097677 {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 A097677 for(n=0,20-r,print1(n!*polcoeff(EGF(x)+x*O(x^n),n),","))
%o A097677 /* -----------------------END---------------------- */
%o A097677 (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)}
%o A097677 (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))))
%Y A097677 Cf. A097663, A097678-A097682.
%K A097677 nonn
%O A097677 0,2
%A A097677 _Paul D. Hanna_, Sep 01 2004