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%I #34 May 02 2022 01:56:27
%S 1,12,366,13800,574650,25335072,1159174464,54435558672,2606102756730,
%T 126634810078920,6226427808402516,309095505195676992,
%U 15466884541698962736,779158533743408851200,39476348002042199114400,2010009672816216740255520
%N Diagonal of 1/(1 - x - y - z - x y - x z - y z).
%C Annihilating differential operator: x*(x+2)*(x+1)*(27*x^2+54*x-1)*Dx^2 + (81*x^4+324*x^3+431*x^2+214*x-2)*Dx + 24*x^3+72*x^2+72*x+24. - _Gheorghe Coserea_, Jul 03 2016
%H Gheorghe Coserea, <a href="/A268550/b268550.txt">Table of n, a(n) for n = 0..310</a>, First 201 terms from Vaclav Kotesovec.
%H A. Bostan, S. Boukraa, J.-M. Maillard, and 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 Steffen Eger, <a href="http://arxiv.org/abs/1511.00622">On the Number of Many-to-Many Alignments of N Sequences</a>, arXiv:1511.00622 [math.CO], 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: 2*n^2*(n-1)*a(n) -(n-1)*(107*n^2-107*n+24)*a(n-1) +12*(-9*n^3+27*n^2-22*n+2)*a(n-2) -3*n*(3*n-5)*(3*n-7)*a(n-3)=0. - _R. J. Mathar_, Mar 10 2016
%F a(n) ~ sqrt(3) * (27 + 6*sqrt(21))^n / (2*Pi*n). - _Vaclav Kotesovec_, Jul 01 2016
%F G.f.: hypergeom([1/12, 5/12], [1], 1728*x^3*(x+2)^3*(27*x^2+54*x-1)/(-1+48*x+24*x^2)^3)/(1-48*x-24*x^2)^(1/4). - _Gheorghe Coserea_, Jul 01 2016
%F 0 = x*(x+2)*(x+1)*(27*x^2+54*x-1)*y'' + (81*x^4+324*x^3+431*x^2+214*x-2)*y' + (24*x^3+72*x^2+72*x+24)*y, where y is g.f. - _Gheorghe Coserea_, Jul 03 2016
%F a(n) = Sum_{i = 0..n} Sum_{k = 0..n-i} Sum_{j = 0..n-k} (n+i+j+k)!/(i!*j!*k!*(n-k-i)!*(n-k-j)!*(2*k+i+j-n)!). Apply [Eger, Theorem 3] to the set of column vectors S = {[1,0,0], [0,1,0], [0,0,1], [1,1,0], [1,0,1], [0,1,1]}. - _Peter Bala_, Apr 28 2022
%p A268550 := proc(n)
%p 1/(1-x-y-z-x*y-x*z-y*z) ;
%p coeftayl(%,x=0,n) ;
%p coeftayl(%,y=0,n) ;
%p coeftayl(%,z=0,n) ;
%p end proc:
%p seq(A268550(n),n=0..20) ; # _R. J. Mathar_, Mar 10 2016
%t gf = Hypergeometric2F1[1/12, 5/12, 1, 1728*x^3*(x + 2)^3*(27*x^2 + 54*x - 1)/(-1 + 48*x + 24*x^2)^3]/(1 - 48*x - 24*x^2)^(1/4);
%t CoefficientList[gf + O[x]^20, 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);
%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 = 16; x = 'x + O('x^N);
%o Vec(hypergeom([1/12, 5/12], [1], 1728*x^3*(x+2)^3*(27*x^2+54*x-1)/(-1+48*x+24*x^2)^3, N)/(1-48*x-24*x^2)^(1/4)) \\ _Gheorghe Coserea_, Jul 03 2016
%Y Cf. A268545-A268555.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Feb 29 2016
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