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%I #17 Mar 19 2023 10:46:09
%S 1,8,138,2960,70090,1756608,45678864,1219013664,33162009210,
%T 915589703600,25578044554348,721420319128704,20509529725235824,
%U 586986330979489280,16895932626393943680,488743896405192037440,14198840150264907505050,414069243091986225102480,12115901803035178006468500
%N Diagonal of the rational function 1/(1 - x - y - z + x y - x z - y z).
%C Annihilating differential operator: x*(x-2)*(11*x+12)*(27*x^2+92*x-3)*Dx^2 + (891*x^4+2132*x^3-689*x^2-4488*x+72)*Dx + 264*x^3+592*x^2+768*x-576.
%H Gheorghe Coserea, <a href="/A274669/b274669.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],-1728*x^4*(27*x^2+92*x-3)*(x-2)^2/(1-32*x+88*x^2)^3)/(1-32*x+88*x^2)^(1/4).
%F 0 = x*(x-2)*(11*x+12)*(27*x^2+92*x-3)*y'' + (891*x^4+2132*x^3-689*x^2-4488*x+72)*y' + (264*x^3+592*x^2+768*x-576)*y, where y is the g.f.
%F D-finite with recurrence 72*(n^2)*a(n) +6*(-363*n^2+341*n-74)*a(n-1) +(-1601*n^2+7316*n-7460)*a(n-2) +2*(371*n^2-1531*n+1550)*a(n-3) +33*(3*n-8)*(3*n-10)*a(n-4)=0. - _R. J. Mathar_, Jul 27 2022
%F a(n) ~ sqrt(35/68 + 32/(17*sqrt(13))) * ((46 + 13*sqrt(13))/3)^n / (Pi*n). - _Vaclav Kotesovec_, Mar 19 2023
%t gf = Hypergeometric2F1[1/12, 5/12, 1, -1728*x^4*(27*x^2 + 92*x - 3)*(x - 2)^2/(1 - 32*x + 88*x^2)^3]/(1 - 32*x + 88*x^2)^(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);
%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 = 20; x = 'x + O('x^N);
%o Vec(hypergeom([1/12, 5/12],[1],-1728*x^4*(27*x^2+92*x-3)*(x-2)^2/(1-32*x+88*x^2)^3, N)/(1-32*x+88*x^2)^(1/4))
%Y Cf. A268545-A268555.
%K nonn
%O 0,2
%A _Gheorghe Coserea_, Jul 05 2016
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