A268553 Diagonal of the rational function 1/((1 - u v - u w - v w) * (1 - x y - x z - y z)).

1, 36, 8100, 2822400, 1200622500, 572679643536, 294230074634496, 159259227403161600, 89595913068008532900, 51926300783585192250000, 30813565377466975498995600, 18639620490164944744006041600, 11456409104219869032980449440000

Offset

0,2

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

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, Eq. (B.21)

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

Formula

Conjecture: n^4*a(n) -9*(3*n-1)^2*(3*n-2)^2*a(n-1) = 0. - R. J. Mathar, Mar 11 2016

From Vaclav Kotesovec, Jul 01 2016: (Start)

a(n) = (3*n)!^2 / (n!)^6.

a(n) ~ 3^(6*n+1) / (4*Pi^2*n^2).

(End)

G.f.: 4F3(1/3,1/3,2/3,2/3; 1,1,1; 729*x). Mathar's conjecture above is true. - Benedict W. J. Irwin, Oct 20 2016

Maple

A268553 := proc(n)
    1/(1-u*v-u*w-v*w)/(1-x*y-x*z-y*z) ;
    coeftayl(%, x=0, 2*n) ;
    coeftayl(%, y=0, 2*n) ;
    coeftayl(%, z=0, 2*n) ;
    coeftayl(%, u=0, 2*n) ;
    coeftayl(%, v=0, 2*n) ;
    coeftayl(%, w=0, 2*n) ;
end proc:
seq(A268553(n), n=0..40) ; # R. J. Mathar, Mar 11 2016

Mathematica

Table[(3*n)!^2 / n!^6, {n, 0, 15}] (* Vaclav Kotesovec, Jul 01 2016 *)
CoefficientList[Series[HypergeometricPFQ[{1/3, 1/3, 2/3, 2/3}, {1, 1, 1}, 729 x], {x, 0, 20}], x] (* Benedict W. J. Irwin, Oct 20 2016 *)

Crossrefs

Sequence in context: A013739 A203052 A054407 * A233171 A058466 A277603

Adjacent sequences: A268550 A268551 A268552 * A268554 A268555 A268556

Keyword

nonn

Author

N. J. A. Sloane, Feb 29 2016

Status

approved