cp's OEIS Frontend

A143522 a(n) = n-fold Dumont operator of x evaluated at x=1, y=1, z=2.

%I A143522 #11 Oct 06 2013 02:39:26
%S A143522 1,2,5,18,93,618,4905,45162,474777,5618322,73895085,1069104258,
%T A143522 16873062453,288485314938,5311769483025,104789840677722,
%U A143522 2205098925335217,49302142664941602,1167150946521879765
%N A143522 a(n) = n-fold Dumont operator of x evaluated at x=1, y=1, z=2.
%C A143522 The Dumont operator: D = y*z*dx + z*x*dy + x*y*dz is used to generate expansions for the Jacobi elliptic functions sn, cn and dn.
%F A143522 E.g.f.: 3/(3*cosh(sqrt(3)*x) - 2*sqrt(3)*sinh(sqrt(3)*x)).
%F A143522 E.g.f.: 2*(3*cosh(sqrt(3)*x) + 2*sqrt(3)*sinh(sqrt(3)*x))/(7 - cosh(2*sqrt(3)*x)).
%F A143522 G.f.: 1/Q(0), where Q(k) = 1 - 2*x*(2*k+1) - x^2*(k+1)^2/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Sep 27 2013
%F A143522 a(n) ~ n!*(2*sqrt(3)/log(7+4*sqrt(3)))^(n+1). - _Vaclav Kotesovec_, Oct 05 2013
%e A143522 Given the Dumont operator: D = y*z*dx + z*x*dy + x*y*dz,
%e A143522 illustrate a(n) = D^n x evaluated at x=1, y=1, z=2:
%e A143522 D^0 x = x --> a(0) = 1;
%e A143522 D^1 x = y*z --> a(1) = 2;
%e A143522 D^2 x = (y^2 + z^2)*x --> a(2) = 5;
%e A143522 D^3 x = 4*z*y*x^2 + (z*y^3 + z^3*y) --> a(3) = 18;
%e A143522 D^4 x = (4*y^2 + 4*z^2)*x^3 + (y^4 + 14*z^2*y^2 + z^4)*x --> a(4) = 93;
%e A143522 D^5 x = 16*z*y*x^4 + (44*z*y^3 + 44*z^3*y)*x^2 + (z*y^5 + 14*z^3*y^3 + z^5*y) --> a(5) = 618.
%t A143522 CoefficientList[Series[3/(3*Cosh[Sqrt[3]*x] - 2*Sqrt[3]*Sinh[Sqrt[3]*x]), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Oct 05 2013 *)
%o A143522 (PARI) {a(n)=local(F=x);if(n>=0,for(i=1,n,F=y*z*deriv(F,x)+z*x*deriv(F,y)+x*y*deriv(F,z))); subst(subst(subst(F,x,1),y,1),z,2)}
%Y A143522 Cf. A143523.
%K A143522 nonn
%O A143522 0,2
%A A143522 _Paul D. Hanna_, Aug 22 2008