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%I #26 Nov 15 2021 11:59:43
%S 1,5,49,593,7921,111965,1641865,24705665,378931585,5898329045,
%T 92893693729,1477015762865,23671209428881,381902943661517,
%U 6196712221450009,101043805539177473,1654726991239056385,27201210101330189477,448652354194417534609,7422254208909904273553
%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*(4*x-5)*(x^3+27*x^2-19*x+1)*Dx^2 + (12*x^4+196*x^3-481*x^2+190*x-5)*Dx + 4*x^3+18*x^2-95*x+25.
%C Also diagonal of rational functions 1/(1 + z + 2*x*y - y*z + x*z - x*y*z), 1/(1 + 2*z + x*y - 2*y*z + x*z - x*y*z), 1/(1 + y + z + 2*x*y + y*z + x*z + x*y*z), 1/(1 + y + 2*z + x*y + 2*y*z + x*z + x*y*z), 1/(1 + x + y - 2*z - y*z - 2*x*z + x*y*z). - _Gheorghe Coserea_, Jul 03 2018
%H Gheorghe Coserea, <a href="/A274671/b274671.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^5*(1-19*x+27*x^2+x^3)/(1-20*x+54*x^2+28*x^3+x^4)^3)/(1-20*x+54*x^2+28*x^3+x^4)^(1/4).
%F 0 = x*(4*x-5)*(x^3+27*x^2-19*x+1)*y'' + (12*x^4+196*x^3-481*x^2+190*x-5)*y' + (4*x^3+18*x^2-95*x+25)*y, where y is the g.f.
%F a(n) = Sum_{k=0..n} binomial(n,k)^2 * binomial(n+k,k) * 2^k. - _Ilya Gutkovskiy_, Nov 15 2021
%F a(n) = hypergeom([-n, -n, n + 1], [1, 1], 2). - _Peter Luschny_, Nov 15 2021
%F Recurrence: n^2*(37*n - 62)*a(n) = (703*n^3 - 1881*n^2 + 1383*n - 330)*a(n-1) - (999*n^3 - 3672*n^2 + 4267*n - 1510)*a(n-2) - (n-2)^2*(37*n - 25)*a(n-3). - _Vaclav Kotesovec_, Nov 15 2021
%t gf = Hypergeometric2F1[1/12, 5/12, 1, 13824*x^5*(1 - 19*x + 27*x^2 + x^3) / (1 - 20*x + 54*x^2 + 28*x^3 + x^4)^3]/(1 - 20*x + 54*x^2 + 28*x^3 + x^4)^(1/4);
%t CoefficientList[gf + O[x]^20, x] (* _Jean-François Alcover_, Dec 01 2017 *)
%t a[n_] := HypergeometricPFQ[{-n, -n, n + 1}, {1, 1}, 2];
%t Table[a[n], {n, 0, 19}] (* _Peter Luschny_, Nov 15 2021 *)
%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^5*(1-19*x+27*x^2+x^3)/(1-20*x+54*x^2+28*x^3+x^4)^3, N)/(1-20*x+54*x^2+28*x^3+x^4)^(1/4))
%o (PARI)
%o diag(expr, N=22, var=variables(expr)) = {
%o my(a = vector(N));
%o for (k = 1, #var, expr = taylor(expr, var[#var - k + 1], N));
%o for (n = 1, N, a[n] = expr;
%o for (k = 1, #var, a[n] = polcoeff(a[n], n-1)));
%o return(a);
%o };
%o diag(1/(1 - x - y - z - x*y + x*z + y*z - x*y*z), 20)
%o \\ test: diag(1/(1 - x - y - z - x*y + x*z + y*z - x*y*z)) == diag(1/(1 + x + y - 2*z - y*z - 2*x*z + x*y*z))
%o \\ _Gheorghe Coserea_, Jul 03 2018
%Y Cf. A268545-A268555.
%K nonn
%O 0,2
%A _Gheorghe Coserea_, Jul 05 2016
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