A276016 Diagonal of (1 - 9 x y)/((1 - 3 y - 2 x + 3 y^2 + 9 x^2 y) * (1 - u - v - w)).

1, 18, 1620, 211680, 32744250, 5578048476, 1011486123648, 191622065057280, 37496545808513220, 7522455177633825000, 1539274868268143806800, 320076654200479057190400, 67449486609121784655852000, 14374016718897406159444968000

Offset

0,2

Comments

"The corresponding (order-three) linear differential operator is not homomorphic to its adjoint, even with an algebraic extension, and its differential Galois group is SL(3,C)." - (see A. Bostan link).

Links

Gheorghe Coserea, Table of n, a(n) for n = 0..33

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. (C.2).

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

Formula

a(n) = [(xyuvw)^n] (1-9*x*y)/((1-3*y-2*x+3*y^2+9*x^2*y) * (1-u-v-w)).

From Vaclav Kotesovec, Dec 03 2017: (Start)

Recurrence: n^3*a(n) = 9*(3*n - 2)^2*(3*n - 1)*a(n-1).

For n > 0, a(n) = 3^(2*n) * Gamma(3*n) * Gamma(n + 1/3) / (Gamma(4/3) * Gamma(n) * Gamma(n+1)^3).

a(n) ~ 3^(5*n + 1/2) / (2*Pi * Gamma(1/3) * n^(5/3)). (End)

Example

1 + 18*x + 1620*x^2 + 211680*x^3 + ...

Maple

diag_coeff := proc(expr, n)
    local var := [seq(indets(expr))], nvar := numelems(var);
    coeftayl(expr, var=[seq(0, i=1..nvar)], [seq(n, i=1..nvar)]);
end proc:
pxy := (1 - 3*y - 2*x + 3*y^2 + 9*x^2*y):
expr := (1 - 9*x*y)/(pxy * (1 - u - v - w)):
[seq(diag_coeff(expr, i), i=0..14)];

Mathematica

f = (1 - 9 x y)/((1 - 3 y - 2 x + 3 y^2 + 9 x^2 y)*(1 - u - v - w));
a[n_] := Fold[SeriesCoefficient[#1, {#2, 0, n}]&, f, {x, y, u, v, w}];
Array[a, 40, 0] (* Jean-François Alcover, Dec 03 2017 *)
Join[{1}, Table[FullSimplify[3^(2*n) * Gamma[3*n] * Gamma[n + 1/3] / (Gamma[4/3] * Gamma[n] * Gamma[n + 1]^3)], {n, 1, 20}]] (* Vaclav Kotesovec, Dec 03 2017 *)

Crossrefs

Cf. A004987, A268549, A268545-A268555.

Sequence in context: A211310 A160307 A003030 * A086366 A086193 A064347

Adjacent sequences: A276013 A276014 A276015 * A276017 A276018 A276019

Keyword

nonn

Author

Gheorghe Coserea, Aug 16 2016

Status

approved