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

1, 2, 5, 18, 93, 618, 4905, 45162, 474777, 5618322, 73895085, 1069104258, 16873062453, 288485314938, 5311769483025, 104789840677722, 2205098925335217, 49302142664941602, 1167150946521879765

Offset: 0

Author

Paul D. Hanna, Aug 22 2008

Keywords

• nonn

Comments

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.

Examples

			Given the Dumont operator: D = y*z*dx + z*x*dy + x*y*dz,
illustrate a(n) = D^n x evaluated at x=1, y=1, z=2:
D^0 x = x --> a(0) = 1;
D^1 x = y*z --> a(1) = 2;
D^2 x = (y^2 + z^2)*x --> a(2) = 5;
D^3 x = 4*z*y*x^2 + (z*y^3 + z^3*y) --> a(3) = 18;
D^4 x = (4*y^2 + 4*z^2)*x^3 + (y^4 + 14*z^2*y^2 + z^4)*x --> a(4) = 93;
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.
		

Crossrefs

Cf. A143523.

Programs

Mathematica

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 *)

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)}

Formula

E.g.f.: 3/(3*cosh(sqrt(3)*x) - 2*sqrt(3)*sinh(sqrt(3)*x)).

E.g.f.: 2*(3*cosh(sqrt(3)*x) + 2*sqrt(3)*sinh(sqrt(3)*x))/(7 - cosh(2*sqrt(3)*x)).

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

a(n) ~ n!*(2*sqrt(3)/log(7+4*sqrt(3)))^(n+1). - Vaclav Kotesovec, Oct 05 2013