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%I #21 Jul 27 2022 10:01:40
%S 1,7,103,1891,38371,824377,18379579,420563731,9810403267,232264240957,
%T 5564072675833,134574852764821,3280845731941519,80522277272406613,
%U 1987608338377888483,49305191067563987731,1228368016027453079587,30719511029184435338053,770839386237255136597501
%N Diagonal of the rational function 1/(1 - x - y - z - x y + x z - x y z).
%C Annihilating differential operator: x*(8*x^2-5*x-24) * (2*x^4-9*x^3+205*x^2-167*x+6)*Dx^2 + (48*x^6-184*x^5+1535*x^4-1186*x^3-13973*x^2+8016*x-144)*Dx + 16*x^5-36*x^4-491*x^3+839*x^2-3840*x+1008.
%C Also diagonal of rational functions 1/(1 + y + 3*z + x*y + y*z + x*z + x*y*z), 1/(1 - 2*y - z - x*y - y*z - x*z - x*y*z), 1/(1 - 3*x + y + z - 2*x*y + y*z - x*z - x*y*z), 1/(1 - 2*x - y - z - x*y + y*z + x*z + x*y*z), 1/(1 - x - y - z + x*y - 2*x*z + x*y*z). - _Gheorghe Coserea_, Jul 03 2018
%H Gheorghe Coserea, <a href="/A274670/b274670.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*(6-167*x+205*x^2-9*x^3+2*x^4)/(1-28*x+78*x^2-4*x^3+x^4)^3)/(1-28*x+78*x^2-4*x^3+x^4)^(1/4).
%F 0 = x*(8*x^2-5*x-24)*(2*x^4-9*x^3+205*x^2-167*x+6)*y'' + (48*x^6-184*x^5+1535*x^4-1186*x^3-13973*x^2+8016*x-144)*y' + (16*x^5-36*x^4-491*x^3+839*x^2-3840*x+1008)*y, where y is the g.f.
%F D-finite with recurrence 144*(n^2)*a(n) +6*(-663*n^2+653*n-158)*a(n-1) +(4037*n^2-6212*n+116)*a(n-2) +(2145*n^2-13829*n+21343)*a(n-3) +(-1637*n^2+13198*n-26109)*a(n-4) +2*(41*n^2-359*n+788)*a(n-5) -16*(n-5)^2*a(n-6)=0. - _R. J. Mathar_, Jul 27 2022
%t gf = Hypergeometric2F1[1/12, 5/12, 1, 1728*x^4*(6 - 167*x + 205*x^2 - 9*x^3 + 2*x^4)/(1 - 28*x + 78*x^2 - 4*x^3 + x^4)^3]/(1 - 28*x + 78*x^2 - 4*x^3 + x^4)^(1/4);
%t CoefficientList[gf + O[x]^20, x] (* _Jean-François Alcover_, Dec 01 2017 *)
%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*(6-167*x+205*x^2-9*x^3+2*x^4)/(1-28*x+78*x^2-4*x^3+x^4)^3, N)/(1-28*x+78*x^2-4*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 - x*y*z), 19)
%o \\ test: diag(1/(1 - x - y - z - x*y + x*z - x*y*z)) == diag(1/(1 - 2*x - y - z - x*y + y*z + 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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