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A143522
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a(n) = n-fold Dumont operator of x evaluated at x=1, y=1, z=2.
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1
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1, 2, 5, 18, 93, 618, 4905, 45162, 474777, 5618322, 73895085, 1069104258, 16873062453, 288485314938, 5311769483025, 104789840677722, 2205098925335217, 49302142664941602, 1167150946521879765
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OFFSET
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0,2
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COMMENTS
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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.
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LINKS
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Table of n, a(n) for n=0..18.
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FORMULA
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E.g.f.: 3/(3*cosh(sqrt(3)*x) - 2*sqrt(3)*sinh(sqrt(3)*x)).
E.g.f.: 2*(3*cosh(sqrt(3)*x) + 2*sqrt(3)*sinh(sqrt(3)*x))/(7 - cosh(2*sqrt(3)*x)).
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EXAMPLE
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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=2:
D^0 x = x --> a(0) = 1;
D^1 x = y*z --> a(1) = 2;
D^2 x = (y^2 + z^2)*x --> a(2) = 5;
D^3 x = 4*z*y*x^2 + (z*y^3 + z^3*y) --> a(3) = 18;
D^4 x = (4*y^2 + 4*z^2)*x^3 + (y^4 + 14*z^2*y^2 + z^4)*x --> a(4) = 93;
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) = 618.
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PROG
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(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, 2)}
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CROSSREFS
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Cf. A143523.
Sequence in context: A099556 A057864 A032273 * A217389 A123310 A058119
Adjacent sequences: A143519 A143520 A143521 * A143523 A143524 A143525
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KEYWORD
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nonn
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AUTHOR
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Paul D. Hanna, Aug 22 2008
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STATUS
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approved
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