0,2

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.

E.g.f.: 2*r*(3-r)*exp(r*x)/(1 - (3-r)^2*exp(2*r*x)) where r=2*sqrt(2).

E.g.f.: G'(x)/G(x) where G(x) = e.g.f. of A080795 (number of minimax trees on n nodes).

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=3:

D^0 x = x --> a(0) = 1;

D^1 x = y*z --> a(1) = 3;

D^2 x = (y^2 + z^2)*x --> a(2) = 10;

D^3 x = 4*z*y*x^2 + (z*y^3 + z^3*y) --> a(3) = 42;

D^4 x = (4*y^2 + 4*z^2)*x^3 + (y^4 + 14*z^2*y^2 + z^4)*x --> a(4) = 248;

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) = 1992.

(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, 3)}

(PARI) {a(n)=local(r=2*sqrt(2)+x*O(x^n)); round(n!*polcoeff(2*r*(3-r)*exp(r*x)/(1-(3-r)^2*exp(2*r*x)), n))}

Cf. A143522, A080795.

Sequence in context: A030964 A030867 A007680 this_sequence A042545 A151084 A151085

Adjacent sequences: A143520 A143521 A143522 this_sequence A143524 A143525 A143526

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Aug 23 2008