login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A275047 Diagonal of the rational function 1/(1-(1+w)(xy + xz + yz)) [even-indexed terms only]. 2
1, 18, 1350, 141120, 17151750, 2272538268, 318430816704, 46404203788800, 6961609406993670, 1068002895589987500, 166779781860762170100, 26422986893371642828800, 4236593267629481817240000, 686167053247777413372681600, 112093956900827388909570240000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Odd-order terms are zero since R(x,y,z,w) = R(-x,-y,-z,w), where R(x,y,z,w) = 1/(1-(1+w)*(x*y + x*z + y*z)).

LINKS

Gheorghe Coserea and Alois P. Heinz, Table of n, a(n) for n = 0..444 (first 34 terms from Gheorghe Coserea)

A. Bostan, S. Boukraa, J.-M. Maillard, J.-A. Weil, Diagonals of rational functions and selected differential Galois groups, arXiv preprint arXiv:1507.03227 [math-ph], 2015.

Jacques-Arthur Weil, Supplementary Material for the Paper "Diagonals of rational functions and selected differential Galois groups"

FORMULA

0 = (-4*x^2+729*x^4)*y'''' + (-20*x+7290*x^3)*y''' + (-16+18063*x^2)*y'' + 10449*x*y' + 576*y, where y = 1 + 18*x^2 + 1350*x^4 + ...

From Vaclav Kotesovec, Aug 03 2016: (Start)

a(n) = (3*n)!^2 / (n!^4 * (2*n)!).

a(n) ~ 3^(6*n+1) / (Pi^(3/2) * n^(3/2) * 2^(2*n+2)).

(End)

G.f.: 4F3(1/3,1/3,2/3,2/3;1/2,1,1;729x/4). - Benedict W. J. Irwin, Aug 05 2016

EXAMPLE

1 + 18*x^2 + 1350*x^4 + 141120*x^6 + ...

MAPLE

a:= proc(n) option remember; `if`(n=0, 1,

      9*(3*n-1)^2*(3*n-2)^2*a(n-1)/((4*n-2)*n^3))

    end:

seq(a(n), n=0..20);  # Alois P. Heinz, Jul 25 2016

MATHEMATICA

Table[(3*n)!^2 / (n!^4*(2*n)!), {n, 0, 20}] (* Vaclav Kotesovec, Aug 03 2016 *)

CoefficientList[Series[HypergeometricPFQ[{1/3, 1/3, 2/3, 2/3}, {1/2, 1, 1}, 729x/4], {x, 0, 10}], x] (* Benedict W. J. Irwin, Aug 05 2016 *)

PROG

my(x='x, y='y, z='z, w='w);

R = 1/(1-(1+w)*(x*y+x*z+y*z));

diag(n, expr, var) = {

  my(a = vector(n));

  for (i = 1, #var, expr = taylor(expr, var[#var - i + 1], n));

  for (k = 1, n, a[k] = expr;

       for (i = 1, #var, a[k] = polcoeff(a[k], k-1)));

  return(a);

};

diag(23, R, [x, y, z, w])

CROSSREFS

Cf. A268545-A268555.

Sequence in context: A252969 A182286 A292609 * A160252 A276015 A210823

Adjacent sequences:  A275044 A275045 A275046 * A275048 A275049 A275050

KEYWORD

nonn

AUTHOR

Gheorghe Coserea, Jul 18 2016

STATUS

approved

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

License Agreements, Terms of Use, Privacy Policy. .

Last modified July 31 22:27 EDT 2021. Contains 346377 sequences. (Running on oeis4.)