A027390 - OEIS

%I #25 Sep 08 2022 08:44:49

%S 1,3,12,56,294,1698,10624,71256,508188,3829652,30340464,251647872,

%T 2177440648,19596586584,182967013056,1768251873632,17653554391056,

%U 181748836240944,1926542292294592,20996051677557888,234959167687868256,2696726674789023008

%N Number of labeled servers of dimension 3.

%H Vincenzo Librandi, <a href="/A027390/b027390.txt">Table of n, a(n) for n = 0..200</a>

%H R. Donaghey, <a href="http://dx.doi.org/10.1016/0097-3165(76)90060-1">Binomial self-inverse sequences and tangent coefficients</a>, J. Combin. Theory, Series A, 21 (1976), 155-163.

%F E.g.f.: exp( (x*(18 + 9*x + 2*x^2))/6 ).

%F a(n) = 3*a(n-1) + 3*(n-1)*a(n-2) + (n-1)*(n-2)*a(n-3). - _Vaclav Kotesovec_, Dec 29 2012 [unless n=0, _Michael Somos_, May 31 2014]

%F a(n) ~ exp(3/2*n^(1/3)+3/2*n^(2/3)-2*n/3-3/2)*n^(2*n/3)/sqrt(3) * (1 + 3/(4*n^(1/3)) + 117/(160*n^(2/3)) - 59/(640*n)). - _Vaclav Kotesovec_, Dec 29 2012

%e G.f. = 1 + 3*x + 12*x^2 + 56*x^3 + 294*x^4 + 1698*x^5 + 10624*x^6 + ...

%t Table[n!*SeriesCoefficient[E^((x*(18 + 9*x + 2*x^2))/6),{x,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Dec 29 2012 *)

%o (Magma) I:=[1, 3, 12, 56]; [n le 3 select I[n] else 3*Self(n-1)+3*(n-2)*Self(n-2)+(n^2 -5*n+6)*Self(n-3): n in [1..25]]; // _Vincenzo Librandi_, Dec 31 2012

%o (PARI) {a(n) = if( n<0, 0, n! * polcoeff( exp((18*x + 9*x^2 + 2*x^3) / 6 + x * O(x^n)), n))}; /* _Michael Somos_, May 31 2014 */

%o (PARI) {a(n) = if( n<1, n==0, 3 * a(n-1) + 3*(n-1) * a(n-2) + (n-1)*(n-2) * a(n-3))}; /* _Michael Somos_, May 31 2014 */

%K nonn

%O 0,2

%A _Olivier Gérard_