OFFSET
0,2
COMMENTS
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)).
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..389 (first 34 terms from Gheorghe Coserea)
A. Bostan, S. Boukraa, J.-M. Maillard, J.-A. Weil, Diagonals of rational functions and selected differential Galois groups, arXiv preprint arXiv:1507.03227 [math-ph], 2015.
FORMULA
a(n) = [(x*y*z*w)^(2n)] 1/(1 - (w*x*z + w*y*z + x*y*z + w*y + x*y + z)).
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).
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
EXAMPLE
1 + 36*x^2 + 5580*x^4 + 1209600*x^6 + ...
MAPLE
a:= proc(n) option remember; `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))
end:
seq(a(n), n=0..20); # Alois P. Heinz, Jul 26 2016
MATHEMATICA
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)];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Sep 07 2017, after Alois P. Heinz *)
PROG
(PARI)
my(x='x, y='y, z='z, w='w);
R = 1/(1-(w*x*z+w*y*z+x*y*z+w*y+x*y+z));
diag(n, expr, var) = {
my(a = vector(n));
for (i = 1, #var, expr = taylor(expr, var[#var - i + 1], n));
for (k = 1, n, a[k] = expr;
for (i = 1, #var, a[k] = polcoeff(a[k], k-1)));
return(a);
};
diag(22, R, [x, y, z, w])
CROSSREFS
KEYWORD
nonn
AUTHOR
Gheorghe Coserea, Jul 19 2016
STATUS
approved