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A143523 a(n) = n-fold Dumont operator of x evaluated at x=y=1, z=3. 2

%I

%S 1,3,10,42,248,1992,19600,222288,2851712,41075328,658359040,

%T 11621260032,223832419328,4669549335552,104894256056320,

%U 2524539033397248,64811332658757632,1767891945806266368,51060500413513400320

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

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

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

%F 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

%F a(n) ~ n! * 2^(5*(n+1)/2) / log(17+12*sqrt(2))^(n+1). - _Vaclav Kotesovec_, Oct 08 2013

%e Given the Dumont operator: D = y*z*dx + z*x*dy + x*y*dz,

%e illustrate a(n) = D^n x evaluated at x=1, y=1, z=3:

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

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

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

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

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

%e 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.

%o (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)}

%o (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))}

%Y Cf. A143522, A080795.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Aug 23 2008

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Last modified January 29 18:12 EST 2023. Contains 359926 sequences. (Running on oeis4.)