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%I #43 Mar 21 2023 13:47:07
%S 1,8,156,3800,102340,2919168,86427264,2626557648,81380484900,
%T 2559296511200,81443222791216,2616761264496288,84749038859067856,
%U 2763262653898544000,90615128199047200800,2986287891921565639200,98841887070519004625700
%N The diagonal of 1/(1 - (y + z + x z + x w + x y w)).
%C From _Gheorghe Coserea_, Jul 03 2016: (Start)
%C Also diagonal of rational function R(x,y,z) = 1/(1 - x - y - z - x*y).
%C Annihilating differential operator: x*(2*x+3)*(16*x^2-71*x+2)*Dx^2 + 2*(32*x^3+x^2-213*x+3)*Dx + 8*x^2+48*x-48.
%C (End)
%H Gheorghe Coserea, <a href="/A268543/b268543.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 S. 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*(17*n-23)*a(n) +(-1207*n^3+2840*n^2-1897*n+360)*a(n-1) + 4*(17*n-6)*(-3+2*n)^2*a(n-2) = 0. - _R. J. Mathar_, Mar 11 2016
%F G.f.: hypergeom([1/12, 5/12], [1], 1728*x^3*(2-71*x+16*x^2)/(1-32*x+16*x^2)^3)*(1-32*x+16*x^2)^(-1/4). - _Gheorghe Coserea_, Jul 01 2016
%F 0 = x*(2*x+3)*(16*x^2-71*x+2)*y'' + 2*(32*x^3+x^2-213*x+3)*y' + (8*x^2+48*x-48)*y, where y is the g.f. - _Gheorghe Coserea_, Jul 03 2016
%F a(n) ~ sqrt(3 + 13/sqrt(17)) * (71+17*sqrt(17))^n / (Pi * n * 2^(2*n + 3/2)). - _Vaclav Kotesovec_, Jul 05 2016
%F From _Peter Bala_, Jan 27 2018: (Start)
%F a(n) = binomial(2*n,n)*Sum_{k = 0..n} binomial(n,k)* binomial(2*n+k,k) (apply Eger, Theorem 3 to the set of column vectors S = {[1,0,0], [0,1,0], [0,0,1], [1,1,0]}). Using this binomial sum, Maple confirms the above recurrence of Mathar.
%F a(n) = A000984(n)*A114496(n). (End)
%p A268543 := proc(n)
%p 1/(1-y-z-x*z-x*w-x*y*w) ;
%p coeftayl(%,x=0,n) ;
%p coeftayl(%,y=0,n) ;
%p coeftayl(%,z=0,n) ;
%p coeftayl(%,w=0,n) ;
%p end proc:
%p seq(A268543(n),n=0..40) ; # _R. J. Mathar_, Mar 11 2016
%p #alternative program
%p with(combinat):
%p seq(binomial(2*n,n)*add(binomial(n,k)*binomial(2*n+k,k), k = 0..n), n = 0..20); # _Peter Bala_, Jan 27 2018
%t CoefficientList[Series[HypergeometricPFQ[{1/12, 5/12}, {1}, 1728*x^3*(2 - 71*x + 16*x^2)/(1 - 32*x + 16*x^2)^3]*(1 - 32*x + 16*x^2)^(-1/4), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Jul 05 2016 *)
%o (PARI)
%o my(x='x, y='y, z='z, w='w);
%o R = 1/(1 - x - y - z - x*y);
%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_sym([1/12,5/12],[1],1728*x^3*(16*x^2-71*x+2)/(16*x^2-32*x+1)^3, N)/(16*x^2-32*x+1)^(1/4)) \\ _Gheorghe Coserea_, Jul 03 2016
%Y Cf. A268545-A268555, A000984, A114496.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Feb 29 2016
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