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Diagonal of the rational function 1/(1 - (w*x*z + w*y*z + x*y*z + w*y + x*y + z)) [even-indexed terms only].
1

%I #29 Oct 27 2023 18:29:15

%S 1,36,5580,1209600,305127900,83936348496,24422566424256,

%T 7391145688692480,2302861234904415900,733755111903173646000,

%U 237987702318837667276080,78313025454309175928186880,26080521003090619899885979200,8773677817145303533293886560000

%N Diagonal of the rational function 1/(1 - (w*x*z + w*y*z + x*y*z + w*y + x*y + z)) [even-indexed terms only].

%C Odd-order terms are zero since R(x,y,z,w) = R(-x,-y,z,-w), where R(x,y,z,w) = 1/(1 - (w*x*z + w*y*z + x*y*z + w*y + x*y + z)).

%H Alois P. Heinz, <a href="/A275050/b275050.txt">Table of n, a(n) for n = 0..389</a> (first 34 terms 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 a(n) = [(x*y*z*w)^(2n)] 1/(1 - (w*x*z + w*y*z + x*y*z + w*y + x*y + z)).

%F 0 = (-50*x^3 + 19341*x^5 - 155898*x^7 - 95256*x^9)*y'''' + (-200*x^2 + 170982*x^4 - 1015686*x^6 - 1143072*x^8)*y''' + (-50*x + 353295*x^3 - 1068420*x^5 - 3513888*x^7)*y'' + (50 + 141777*x^2 + 282540*x^4 - 2709504*x^6)*y' + (66816*x^3 - 254016*x^5)*y, where y(x) = A(x^2).

%F a(n) ~ sqrt(1 + 2*sqrt(3/11)) * (189 + 33*sqrt(33))^n / (2*Pi^(3/2)*n^(3/2)). - _Vaclav Kotesovec_, Aug 03 2016

%e 1 + 36*x^2 + 5580*x^4 + 1209600*x^6 + ...

%p a:= proc(n) option remember; `if`(n<2, 36^n, (9*(n-1)*

%p (2*n-1)*(693*n^3-1554*n^2+989*n-160)*a(n-1)+

%p (6*(33*n-8))*(3*n-4)*(2*n-1)*(2*n-3)*(3*n-5)*

%p a(n-2)) / ((n-1)*(33*n-41)*n^3))

%p end:

%p seq(a(n), n=0..20); # _Alois P. Heinz_, Jul 26 2016

%t a[n_] := a[n] = If[n < 2, 36^n, (9*(n - 1)*(2*n - 1)*(693*n^3 - 1554*n^2 + 989*n - 160)*a[n - 1] + (6*(33*n - 8))*(3*n - 4)*(2*n - 1)*(2*n - 3)*(3*n - 5)*a[n - 2])/((n - 1)*(33*n - 41)*n^3)];

%t Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Sep 07 2017, after _Alois P. Heinz_ *)

%o (PARI)

%o my(x='x, y='y, z='z, w='w);

%o R = 1/(1-(w*x*z+w*y*z+x*y*z+w*y+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(22, R, [x,y,z,w])

%Y Cf. A268545-A268555.

%K nonn

%O 0,2

%A _Gheorghe Coserea_, Jul 19 2016