A274674 - OEIS

%I #15 Dec 02 2017 20:48:57

%S 1,1,7,37,211,1351,8611,57037,383587,2615851,18052057,125693107,

%T 882033439,6229779739,44246291467,315774707437,2263120500067,

%U 16279948902259,117498622706269,850541100418807,6173221388110861,44912998208539561,327476893004792197,2392516335780421627

%N Diagonal of the rational function 1/(1 - x - x y - x z - y z + x y z).

%C Annihilating differential operator: x*(6*x^2-4*x-5)*(2*x^4-64*x^3-27*x^2-3*x+1)*Dx^2 + (36*x^6-800*x^5+556*x^4+1496*x^3+411*x^2+30*x-5)*Dx + 12*x^5-100*x^4+256*x^3+540*x^2+105*x+5.

%H Gheorghe Coserea, <a href="/A274674/b274674.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^5*(1-3*x-27*x^2-64*x^3+2*x^4)/(1-4*x-18*x^2-28*x^3+x^4)^3)/(1-4*x-18*x^2-28*x^3+x^4)^(1/4).

%F 0 = x*(6*x^2-4*x-5)*(2*x^4-64*x^3-27*x^2-3*x+1)*y'' + (36*x^6-800*x^5+556*x^4+1496*x^3+411*x^2+30*x-5)*y' + (12*x^5-100*x^4+256*x^3+540*x^2+105*x+5)*y, where y is the g.f.

%F Recurrence: n^2*(517*n^2 - 2249*n + 2322)*a(n) = (1551*n^4 - 8298*n^3 + 13910*n^2 - 8103*n + 1530)*a(n-1) + (13959*n^4 - 88641*n^3 + 196637*n^2 - 178937*n + 54690)*a(n-2) + 2*(16544*n^4 - 121600*n^3 + 316309*n^2 - 336617*n + 117690)*a(n-3) - 2*(n-3)^2*(517*n^2 - 1215*n + 590)*a(n-4). - _Vaclav Kotesovec_, Jul 07 2016

%t gf = Hypergeometric2F1[1/12, 5/12, 1, 1728*x^5*(1 - 3*x - 27*x^2 - 64*x^3 + 2*x^4)/(1 - 4*x - 18*x^2 - 28*x^3 + x^4)^3]/(1 - 4*x - 18*x^2 - 28*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 - x*y - x*z - 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 = 24; x = 'x + O('x^N);

%o Vec(hypergeom([1/12, 5/12],[1],1728*x^5*(1-3*x-27*x^2-64*x^3+2*x^4)/(1-4*x-18*x^2-28*x^3+x^4)^3, N)/(1-4*x-18*x^2-28*x^3+x^4)^(1/4))

%Y Cf. A268545-A268555.

%K nonn

%O 0,3

%A _Gheorghe Coserea_, Jul 06 2016