OFFSET
0,2
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.
FORMULA
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) is the e.g.f. of A080795 (number of minimax trees on n nodes).
G.f.: 1/Q(0), where Q(k) = 1 - 3*x*(2*k+1) - x^2*(k+1)^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Sep 28 2013
a(n) ~ n! * 2^(5*(n+1)/2) / log(17+12*sqrt(2))^(n+1). - Vaclav Kotesovec, Oct 08 2013
EXAMPLE
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.
PROG
(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))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 23 2008
STATUS
approved