|
%I #13 Mar 20 2023 04:16:07
%S 1,13,589,37501,2776861,224127793,19128964429,1697822272957,
%T 155124241235293,14493082279323913,1378280656022778289,
%U 132975844343348756257,12983655880217911846621,1280541892692200972993809,127387054518359023378891069,12766850683487700784950948541,1287829822333113383109436556893
%N Diagonal of the rational function 1/(1-(wxyz + wy + wz + x + y + z)).
%H Vaclav Kotesovec, <a href="/A275045/b275045.txt">Table of n, a(n) for n = 0..488</a> (terms 0..33 from Gheorghe Coserea)
%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 0 = (-x^2+108*x^3+330*x^4-4*x^5-441*x^6+12*x^7-4*x^8)*y''' + (-3*x+498*x^2+660*x^3-1350*x^4-1281*x^5+42*x^6-24*x^7)*y'' + (-1+366*x-795*x^2-1168*x^3-330*x^4+12*x^5-28*x^6)*y' + (13-215*x+208*x^2+2*x^3-4*x^4-4*x^5)*y, where y is g.f.
%F Recurrence: n^3*(3*n - 7)*(3*n - 5)*(3*n - 4)*a(n) = (3*n - 7)*(3*n - 2)*(336*n^4 - 1288*n^3 + 1638*n^2 - 783*n + 123)*a(n-1) - 3*(3*n - 5)*(342*n^5 - 2280*n^4 + 5579*n^3 - 6102*n^2 + 2891*n - 466)*a(n-2) + 2*(3*n - 1)*(18*n^5 - 171*n^4 + 622*n^3 - 1062*n^2 + 825*n - 222)*a(n-3) - (n-3)^3*(3*n - 4)*(3*n - 2)*(3*n - 1)*a(n-4). - _Vaclav Kotesovec_, Mar 20 2023
%o (PARI)
%o my(x='x, y='y, z='z, w='w);
%o R = 1/(1-(w*x*y*z+w*y+w*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(12, R, [x,y,z,w])
%Y Cf. A268545-A268555.
%K nonn
%O 0,2
%A _Gheorghe Coserea_, Jul 15 2016
| 