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%I #27 Mar 25 2023 08:19:00
%S 1,11,283,9155,327811,12436541,489807991,19803209843,816330309475,
%T 34156900690841,1446223566321733,61826502242685653,
%U 2664286789334520559,115586782462237980905,5043474229642670729743,221159937117980575239635,9740104064284459778657635,430601976748346102416423025
%N Diagonal of the rational function 1/(1 - x - y - z - x y - y z - x y z).
%C Annihilating differential operator: x*(x-2)*(5*x+2)*(3*x^2-47*x+1)*Dx^2 + (45*x^4-506*x^3+157*x^2+380*x-4)*Dx + 15*x^3-63*x^2-136*x+44.
%H Gheorghe Coserea, <a href="/A274780/b274780.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 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 G.f.: hypergeom([1/12, 5/12], [1],1728*(3*x^2-47*x+1)*(x-2)^2*x^3/(x^4-20*x^3+78*x^2-44*x+1)^3)/(x^4-20*x^3+78*x^2-44*x+1)^(1/4).
%F 0 = x*(x-2)*(5*x+2)*(3*x^2-47*x+1)*y'' + (45*x^4-506*x^3+157*x^2+380*x-4)*y' + (15*x^3-63*x^2-136*x+44)*y, where y is the g.f.
%F Recurrence: 2*n^2*(39*n - 53)*a(n) = (3705*n^3 - 8740*n^2 + 5823*n - 1096)*a(n-1) - (2067*n^3 - 6943*n^2 + 6915*n - 1664)*a(n-2) + 3*(n-2)^2*(39*n - 14)*a(n-3). - _Vaclav Kotesovec_, Jul 07 2016
%F a(n) ~ sqrt(29/2 + 103/(2*sqrt(13))) * ((47+13*sqrt(13))/2)^n / (6*Pi*n). - _Vaclav Kotesovec_, Jul 07 2016
%F a(n) = Sum_{j = 0..n} Sum_{i = 0..j} Sum_{k = 0..2*n+i} C(2*j,j)*C(j,i)*C(n+j,2*j)*C(k,n+j)*C(n-j,2*n+i-k) (apply Eger, Theorem 3 to the set of column vectors S = {[1,0,0], [0,1,0], [1,1,0], [0,0,1], [0,1,1], [1,1,1]}). - _Peter Bala_, Jan 26 2018
%p with(combinat): seq(add(add(add(binomial(2*j,j)*binomial(j,i)*binomial(n + j,2*j)*binomial(k,n + j)*binomial(n - j,2*n + i - k), k = 0..2*n+i), i = 0..j), j = 0..n), n = 0..20); # _Peter Bala_, Jan 26 2018
%t gf = Hypergeometric2F1[1/12, 5/12, 1, 1728*(3*x^2 - 47*x + 1)*(x - 2)^2*x^3 / (x^4 - 20*x^3 + 78*x^2 - 44*x + 1)^3]/(x^4 - 20*x^3 + 78*x^2 - 44*x + 1)^(1/4);
%t CoefficientList[gf + O[x]^20, x] (* _Jean-François Alcover_, Dec 01 2017 *)
%o (PARI)
%o my(x='x, y='y, z='z);
%o R = 1/(1 - x - y - z - x*y - 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_sym([1/12, 5/12], [1],1728*(3*x^2-47*x+1)*(x-2)^2*x^3/(x^4-20*x^3+78*x^2-44*x+1)^3, N)/(x^4-20*x^3+78*x^2-44*x+1)^(1/4))
%Y Cf. A268545-A268555.
%K nonn,easy
%O 0,2
%A _Gheorghe Coserea_, Jul 06 2016