A143523 a(n) = n-fold Dumont operator of x evaluated at x=y=1, z=3.

1, 3, 10, 42, 248, 1992, 19600, 222288, 2851712, 41075328, 658359040, 11621260032, 223832419328, 4669549335552, 104894256056320, 2524539033397248, 64811332658757632, 1767891945806266368, 51060500413513400320

Offset: 0

Author

Paul D. Hanna, Aug 23 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=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.
		

Crossrefs

Cf. A143522, A080795.

Programs

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))}

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