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%I #23 Dec 03 2017 02:09:17
%S 1,11,325,11711,465601,19590491,855266581,38319499775,1750193256961,
%T 81131090245931,3805404745303525,180207832513958975,
%U 8601942203526345025,413358969518738106875,19977566733574388828725,970297391859524593324031,47330511448436249282088961
%N Diagonal of 1/(1 - x + y + z + x y + x z - y z + x y z).
%H Vaclav Kotesovec, <a href="/A268551/b268551.txt">Table of n, a(n) for n = 0..170</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, 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 Conjecture: n^2*(n-2)*a(n) +(-50*n^3+149*n^2-109*n+21)*a(n-1) -(2*n-3) *(51*n^2-153*n+91)*a(n-2) +(-50*n^3+301*n^2-565*n+315)*a(n-3) +(n-1)*(n-3)^2*a(n-4)=0. - _R. J. Mathar_, Mar 10 2016
%F a(n) ~ (1+sqrt(3))^(6*n+3) / (3*Pi*n*2^(3*n+3)). - _Vaclav Kotesovec_, Jul 01 2016
%F From _Gheorghe Coserea_, Jul 07 2016, (Start)
%F G.f.: hypergeom([1/12, 5/12],[1],13824*x^3*(x^2-52*x+1)/(x^2-46*x+1)^3/(x+1)^2)/((x^2-46*x+1)*(x+1)^2)^(1/4).
%F 0 = x*(x-1)*(x^2-52*x+1)*(x+1)^2*y'' + (x+1)*(3*x^4-106*x^3+102*x^2+102*x-1)*y' + (x^4-12*x^3+32*x^2+68*x+11)*y, where y is g.f.
%F Annihilating differential operator: x*(x-1)*(x^2-52*x+1)*(x+1)^2*Dx^2 + (x+1)*(3*x^4-106*x^3+102*x^2+102*x-1)*Dx + x^4-12*x^3+32*x^2+68*x+11.
%F (End)
%p A268551 := proc(n)
%p 1/(1-x+y+z+x*y+x*z-y*z+x*y*z) ;
%p coeftayl(%,x=0,n) ;
%p coeftayl(%,y=0,n) ;
%p coeftayl(%,z=0,n) ;
%p end proc:
%p seq(A268551(n),n=0..40) ; # _R. J. Mathar_, Mar 10 2016
%t gf = Hypergeometric2F1[1/12, 5/12, 1, 13824*x^3*(x^2 - 52*x + 1)/(x^2 - 46*x + 1)^3/(x + 1)^2]/((x^2 - 46*x + 1)*(x + 1)^2)^(1/4);
%t CoefficientList[gf + O[x]^40, x] (* _Jean-François Alcover_, Dec 03 2017, after _Gheorghe Coserea_ *)
%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 = 20; x = 'x + O('x^N);
%o Vec(hypergeom([1/12, 5/12],[1],13824*x^3*(x^2-52*x+1)/(x^2-46*x+1)^3/(x+1)^2, N)/((x^2-46*x+1)*(x+1)^2)^(1/4)) \\ _Gheorghe Coserea_, Jul 06 2016
%Y Cf. A268545-A268555.
%K nonn
%O 0,2
%A _N. J. A. Sloane_, Feb 29 2016
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