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A097679 E.g.f.: (1/(1-x^4))*exp( 4*Sum_{i>=0} x^(4*i+1)/(4*i+1) ) for an order-4 linear recurrence with varying coefficients.

%I A097679 #27 Feb 16 2025 08:32:54
%S A097679 1,4,16,64,280,1600,12160,102400,880000,8358400,94720000,1189888000,
%T A097679 15213952000,204285952000,3092697088000,51351519232000,
%U A097679 869951500288000,15148619579392000,287722152460288000,5927812334878720000
%N A097679 E.g.f.: (1/(1-x^4))*exp( 4*Sum_{i>=0} x^(4*i+1)/(4*i+1) ) for an order-4 linear recurrence with varying coefficients.
%C A097679 Lim_{n->inf} n*n!/a(n) = 4*c = 0.4157591527... where c = 4*exp(psi(1/4)+EulerGamma) = 0.1039397881...(A097665) and EulerGamma is the Euler-Mascheroni constant (A001620) and psi() is the Digamma function (see Mathworld link).
%D A097679 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 A097679 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 A097679 Benoit Cloitre, <a href="/A097679/a097679.pdf">On a generalization of Euler-Gauss formula for the Gamma function</a>, preprint 2004.
%H A097679 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 A097679 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DigammaFunction.html">Digamma Function</a>.
%F A097679 For n>=4: a(n) = 4*a(n-1) + n!/(n-4)!*a(n-4); for n<4: a(n)=4^n.
%F A097679 E.g.f.: (1+x)/(1-x^4)/(1-x)*exp(2*atan(x)).
%e A097679 The sequence {1, 4, 16/2!, 64/3!, 280/4!, 1600/5!, 12160/6!, 102400/7!,...} is generated by a recursion described by _Benoit Cloitre_'s generalized Euler-Gauss formula for the Gamma function (see Cloitre link).
%t A097679 Range[0, 20]! CoefficientList[ Series[ E^(4Sum[x^(4k + 1)/(4k + 1), {k, 0, 150}])/(1 - x^4), {x, 0, 20}], x] (* _Robert G. Wilson v_, Sep 03 2004 *)
%o A097679 The following PARI code generates this sequence and demonstrates the general recursion with the asymptotic limit and e.g.f.:
%o A097679 /* Define Cloitre's recursion: */
%o A097679 z=[1,0,0,0]; r=4; s=4; zt=sum(i=1,r,z[i])
%o A097679 {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 A097679 /* The following tends to a limit (slowly): */
%o A097679 for(n=r,20,print(if(w(n)==0,0,n^zt/w(n))*1.0,","))
%o A097679 /* This is the exact value of the limit: */
%o A097679 {s^(zt+1)*gamma(zt+1)*exp(sum(k=1,r,z[k]*(psi(k/s)+Euler)))}
%o A097679 /* Print terms w(n) multiplied by (n-r)! for e.g.f. */
%o A097679 for(n=r,20,print1((n-r)!*w(n),","))
%o A097679 /* Compare to terms generated by e.g.f.: */
%o A097679 {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 A097679 for(n=0,20-r,print1(n!*polcoeff(EGF(x)+x*O(x^n),n),","))
%o A097679 /* -----------------------END---------------------- */
%o A097679 (PARI) {a(n)=n!*polcoeff(1/(1-x^4)*exp(4*sum(i=0,n,x^(4*i+1)/(4*i+1)))+x*O(x^n),n)}
%o A097679 (PARI) a(n)=if(n<0,0,if(n==0,1,4*a(n-1)+if(n<4,0,n!/(n-4)!*a(n-4))))
%o A097679 (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!((1+x)/(1-x^4)/(1-x)*Exp(2*Arctan(x)))); [Factorial(n-1)*b[n]: n in [1..m]]; // _G. C. Greubel_, Aug 29 2018
%Y A097679 Cf. A097665, A097677, A097678, A097680, A097681, A097682.
%K A097679 nonn
%O A097679 0,2
%A A097679 _Paul D. Hanna_, Sep 01 2004