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 A143523 a(n) = n-fold Dumont operator of x evaluated at x=y=1, z=3. 2
 1, 3, 10, 42, 248, 1992, 19600, 222288, 2851712, 41075328, 658359040, 11621260032, 223832419328, 4669549335552, 104894256056320, 2524539033397248, 64811332658757632, 1767891945806266368, 51060500413513400320 (list; graph; refs; listen; history; text; internal format)
 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. LINKS 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 Cf. A143522, A080795. Sequence in context: A334270 A185621 A190657 * A288813 A306237 A042545 Adjacent sequences: A143520 A143521 A143522 * A143524 A143525 A143526 KEYWORD nonn AUTHOR Paul D. Hanna, Aug 23 2008 STATUS approved

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Last modified December 4 09:27 EST 2022. Contains 358556 sequences. (Running on oeis4.)