|
Search: id:A143522
|
|
|
| A143522 |
|
a(n) = n-fold Dumont operator of x evaluated at x=1, y=1, z=2. |
|
+0
2
|
|
| 1, 2, 5, 18, 93, 618, 4905, 45162, 474777, 5618322, 73895085, 1069104258, 16873062453, 288485314938, 5311769483025, 104789840677722, 2205098925335217, 49302142664941602, 1167150946521879765
(list; graph; listen)
|
|
|
OFFSET
|
0,2
|
|
|
COMMENT
|
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.: 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)).
|
|
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=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.
|
|
PROGRAM
|
(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)}
|
|
CROSSREFS
|
Cf. A143523.
Sequence in context: A099556 A057864 A032273 this_sequence A123310 A058119 A075634
Adjacent sequences: A143519 A143520 A143521 this_sequence A143523 A143524 A143525
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
Paul D. Hanna (pauldhanna(AT)juno.com), Aug 22 2008
|
|
|
Search completed in 0.002 seconds
|
