|
%I #22 Dec 03 2017 17:08:00
%S 1,12,864,100800,14112000,2139830784,338341183488,54913641209856,
%T 9080061146956800,1523231914913280000,258557709598427086848,
%U 44324863067728222027776,7663322563977594870300672,1334677098876385703362560000,233951210561895726160281600000
%N Diagonal of (1 - 9 x y) / ((1 - 3 y - 2 x + 3 y^2 + 8 x^2 y) * (1 - u - z) * (1 - v - w)).
%C "The corresponding (order-five) linear differential operator is not homomorphic to its adjoint, even with an algebraic extension, and its differential Galois group is SL(5,C)." (see A. Bostan link).
%H Gheorghe Coserea, <a href="/A276013/b276013.txt">Table of n, a(n) for n = 0..33</a>
%H A. Bostan, S. Boukraa, J.-M. Maillard, J.-A. Weil, <a href="http://arxiv.org/abs/1507.03227">Diagonals of rational functions and selected differential Galois groups</a>, arXiv preprint arXiv:1507.03227 [math-ph], 2015, Eq. (36).
%H Jacques-Arthur Weil, <a href="http://www.unilim.fr/pages_perso/jacques-arthur.weil/diagonals/">Supplementary Material for the Paper "Diagonals of rational functions and selected differential Galois groups"</a>
%F a(n) = [(xyzuvw)^n] (1-9*x*y)/((1 - 3*y - 2*x + 3*y^2 + 8*x^2*y) * (1-u-z) * (1-v-w)).
%F From _Vaclav Kotesovec_, Dec 03 2017: (Start)
%F Recurrence: (n-1)^2*n^3*(3*n - 5)*a(n) = 24*(n-1)^2*(2*n - 1)^2*(3*n - 4)*(3*n - 2)*a(n-1) - 384*(2*n - 3)^2*(2*n - 1)^2*(3*n - 5)*(3*n - 2)*a(n-2).
%F a(n) ~ Gamma(1/3) * 2^(6*n - 7/3) * 3^(n + 1/2) / (Pi^2 * n^(4/3)). (End)
%e 1 + 12*x + 864*x^2 + 100800*x^3 + ...
%p diag_coeff := proc(expr, n)
%p local var := [seq(indets(expr))], nvar := numelems(var);
%p coeftayl(expr, var=[seq(0, i=1..nvar)], [seq(n, i=1..nvar)])
%p end proc:
%p pxy := (1 - 3*y - 2*x + 3*y^2 + 9*x^2*y):
%p expr := (1 - 9*x*y)/(pxy * (1-u-z-u*z) * (1-v-w)):
%p [seq(diag_coeff(expr, i), i=0..14)];
%t f = (1 - 9 x y)/((1 - 3y - 2x + 3 y^2 + 8 x^2 y)*(1 - u - z)*(1 - v - w));
%t a[n_] := Fold[SeriesCoefficient[#1, {#2, 0, n}] &, f, {x, y, z, u, v, w}];
%t Array[a, 40, 0] (* _Jean-François Alcover_, Dec 03 2017 *)
%Y Cf. A004987, A268549, A268545-A268555.
%K nonn
%O 0,2
%A _Gheorghe Coserea_, Aug 16 2016
| 