cp's OEIS Frontend

A276014 Diagonal of (1 - 9 x y) / ((1 - 3 y - 2 x + 3 y^2 + 9 x^2 y) * (1 - u - v - z - w)).

%I A276014 #24 Dec 03 2017 08:19:27
%S A276014 1,72,45360,46569600,59594535000,86482063571904,136141986298526208,
%T A276014 226888189910421811200,394399917777684601926000,
%U A276014 708188604075430924446000000,1304782547573305224852017990400,2454776409299366206456800694732800,4699106882676505497505898579906736000,9127695522416954472516114289988092800000
%N A276014 Diagonal of (1 - 9 x y) / ((1 - 3 y - 2 x + 3 y^2 + 9 x^2 y) * (1 - u - v - z - w)).
%C A276014 "The corresponding (order-four) linear differential operator is not homomorphic to its adjoint, even with an algebraic extension, and its differential Galois group is SL(4,C)." (see A. Bostan link).
%H A276014 Gheorghe Coserea, <a href="/A276014/b276014.txt">Table of n, a(n) for n = 0..33</a>
%H A276014 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. (37).
%H A276014 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 A276014 a(n) = [(xyzuvw)^n] (1-9*x*y)/((1-3*y-2*x + 3*y^2 + 9*x^2*y) * (1-u-v-z-w)).
%F A276014 From _Vaclav Kotesovec_, Dec 03 2017: (Start)
%F A276014 Recurrence: n^4*a(n) = 24*(2*n - 1)*(3*n - 2)*(4*n - 3)*(4*n - 1)*a(n-1).
%F A276014 For n > 0, a(n) = 4 * 9^n * Gamma(4*n) * Gamma(n + 1/3) / (Gamma(1/3) * Gamma(n) * Gamma(n+1)^4).
%F A276014 a(n) ~ 2^(8*n - 1/2) * 3^(2*n) / (Pi^(3/2) * Gamma(1/3) * n^(13/6)). (End)
%e A276014 1 + 72*x + 45360*x^2 + 46569600*x^3 + ...
%p A276014 diag_coeff := proc(expr, n)
%p A276014     local var := [seq(indets(expr))], nvar := numelems(var);
%p A276014     coeftayl(expr, var=[seq(0, i=1..nvar)], [seq(n, i=1..nvar)]);
%p A276014 end proc:
%p A276014 expr := (1-9*x*y)/((1-3*y-2*x + 3*y^2 + 9*x^2*y) * (1-u-v-z-w)):
%p A276014 [seq(diag_coeff(expr, i), i=0..14)];
%t A276014 f = (1 - 9 x y)/((1 - 3 y - 2 x + 3 y^2 + 9 x^2 y)*(1 - u - v - z - w));
%t A276014 a[n_] := Fold[SeriesCoefficient[#1, {#2, 0, n}] &, f, {x, y, z, u, v, w}];
%t A276014 Array[a, 40, 0] (* _Jean-François Alcover_, Dec 03 2017 *)
%t A276014 Join[{1}, Table[FullSimplify[(4 * 9^n * Gamma[4*n] * Gamma[1/3 + n]) / (Gamma[1/3] * Gamma[n] * Gamma[1 + n]^4)], {n, 1, 20}]] (* _Vaclav Kotesovec_, Dec 03 2017 *)
%Y A276014 Cf. A004987, A268549, A268545-A268555.
%K A276014 nonn
%O A276014 0,2
%A A276014 _Gheorghe Coserea_, Aug 16 2016