A274672 - OEIS

%I #14 Jul 27 2022 10:04:15

%S 1,9,211,6093,194211,6552759,229425631,8244126669,302001325987,

%T 11228376095379,422418832551981,16044756006730779,614293828071090831,

%U 23677157141279202843,917844665558530089591,35756691970868828784813,1399009670335978828299747,54945874078445147692928043

%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)*(8*x^2-x-6)*(2*x^3-29*x^2-41*x+1)*Dx^2 + (48*x^6-408*x^5-777*x^4+798*x^3+1847*x^2+972*x-12)*Dx + 16*x^5-52*x^4+141*x^3+529*x^2+408*x+108

%H Gheorghe Coserea, <a href="/A274672/b274672.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^3*(2*x^3-29*x^2-41*x+1)*(x+2)^2/(1-36*x-34*x^2-12*x^3+x^4)^3)/(1-36*x-34*x^2-12*x^3+x^4)^(1/4).

%F 0 = x*(x+2)*(8*x^2-x-6)*(2*x^3-29*x^2-41*x+1)*y'' + (48*x^6-408*x^5-777*x^4+798*x^3+1847*x^2+972*x-12)*y' + (16*x^5-52*x^4+141*x^3+529*x^2+408*x+108)*y, where y is the g.f.

%F D-finite with recurrence 12*(n^2)*a(n) +4*(-121*n^2+120*n-26)*a(n-1) +(-691*n^2+1608*n-860)*a(n-2) +(399*n^2-3591*n+6653)*a(n-3) +(779*n^2-6234*n+12331)*a(n-4) +2*(101*n^2-907*n+2036)*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^3*(2*x^3 - 29*x^2 - 41*x + 1)*(x + 2)^2/(1 - 36*x - 34*x^2 - 12*x^3 + x^4)^3]/(1 - 36*x - 34*x^2 - 12*x^3 + x^4)^(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([1/12, 5/12],[1],1728*x^3*(2*x^3-29*x^2-41*x+1)*(x+2)^2/(1-36*x-34*x^2-12*x^3+x^4)^3, N)/(1-36*x-34*x^2-12*x^3+x^4)^(1/4))

%Y Cf. A268545-A268555.

%K nonn

%O 0,2

%A _Gheorghe Coserea_, Jul 05 2016