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%I #25 Jan 18 2020 18:21:27
%S 1,7,109,2095,44401,995647,23161909,552919423,13454515585,
%T 332268466327,8302478659069,209447296631503,5325782947464721,
%U 136330694520639535,3509805380065157989,90806156097601965055,2359490223343888886785,61541525049445532797735,1610570872210945422212365
%N Diagonal of the rational function 1/(1 - x - y - z - x y + x z - y z + x y z).
%C Annihilating differential operator: x*(5*x+4)*(x^3+19*x^2+27*x-1)*Dx^2 + (15*x^4+206*x^3+363*x^2+216*x-4)*Dx + 5*x^3+33*x^2+36*x+28.
%H Gheorghe Coserea, <a href="/A274668/b274668.txt">Table of n, a(n) for n = 0..310</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.
%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 G.f.: hypergeom([1/12, 5/12],[1],13824*x^4*(1-27*x-19*x^2-x^3)/(1-28*x+54*x^2+20*x^3+x^4)^3)/(1-28*x+54*x^2+20*x^3+x^4)^(1/4).
%F 0 = x*(5*x+4)*(x^3+19*x^2+27*x-1)*y'' + (15*x^4+206*x^3+363*x^2+216*x-4)*y' + (5*x^3+33*x^2+36*x+28)*y, where y is the g.f.
%F From _Peter Bala_, Jan 15 2020: (Start)
%F a(n) = Sum_{0 <= j, k <= n} (-1)^(n+k)*C(n,k)*C(n,j)*C(n+k,k)*C(n+k+j,k+j). Cf. A001850 and A126086.
%F n^2*(37*n - 49)*a(n) = (999*n^3 - 2322*n^2 + 1567*n - 328)*a(n-1) + (703*n^3 - 2337*n^2 + 2295*n - 536)*a(n-2) + (n - 2)^2*(37*n - 12)*a(n-3). (End)
%t gf = Hypergeometric2F1[1/12, 5/12, 1, 13824*x^4*(1 - 27*x - 19*x^2 - x^3) / (1 - 28*x + 54*x^2 + 20*x^3 + x^4)^3]/(1 - 28*x + 54*x^2 + 20*x^3 + x^4)^(1/4);
%t CoefficientList[gf + O[x]^20, x] (* _Jean-François Alcover_, Dec 01 2017 *)
%o (PARI)
%o my(x='x, y='y, z='z);
%o R = 1/(1 - x - y - z - x*y + x*z - y*z + x*y*z);
%o diag(n, expr, var) = {
%o my(a = vector(n));
%o for (i = 1, #var, expr = taylor(expr, var[#var - i + 1], n));
%o for (k = 1, n, a[k] = expr;
%o for (i = 1, #var, a[k] = polcoeff(a[k], k-1)));
%o return(a);
%o };
%o diag(10, R, [x, y, z])
%o (PARI) \\ system("wget http://www.jjj.de/pari/hypergeom.gpi");
%o read("hypergeom.gpi");
%o N = 21; x = 'x + O('x^N);
%o Vec(hypergeom([1/12, 5/12],[1],13824*x^4*(1-27*x-19*x^2-x^3)/(1-28*x+54*x^2+20*x^3+x^4)^3, N)/(1-28*x+54*x^2+20*x^3+x^4)^(1/4))
%Y Cf. A268545-A268555.
%K nonn
%O 0,2
%A _Gheorghe Coserea_, Jul 05 2016
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