login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

Table of n, a(n) for n=0..18.

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

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