%I #20 Aug 04 2016 05:54:26

%S 1,616,947175,1812651820,3833011883965,8582819380142616,

%T 19946071353510410136,47578122531207001944168,

%U 115702070514540009854741415,285583642613093627090885877280,713269435359072253352128013072035,1798640871397075848520426334656821360,4571867984920021228226177292547960261132

%N Diagonal of the rational function 1/(1 - 3x - 5y - 7z + xy + 2yz^2 + 3x^2z^2).

%H Gheorghe Coserea and Vaclav Kotesovec, <a href="/A275048/b275048.txt">Table of n, a(n) for n = 0..134</a> (terms 0..55 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>

%H S. Boukraa, S. Hassani, J-M. Maillard and J-A. Weil, <a href="https://arxiv.org/abs/1311.2470">Differential algebra on lattice Green functions and Calabi-Yau operators (unabridged version)</a>, arXiv:1311.2470 [math-ph], 2013.

%F a(n) = [(xyz)^n] 1/(1-3*x-5*y-7*z+x*y+2*y*z^2+3*x^2*z^2).

%F a(n) ~ c*d^n/n, where d = 2768.26905153125473648033608782909246512063674470098..., c = 0.275933667674847170980793689... . - _Vaclav Kotesovec_, Aug 03 2016

%p a:=proc(n) 1/(1-3*x-5*y-7*z+x*y+2*y*z^2+3*x^2*z^2); coeftayl(%, x=0, n); coeftayl(%, y=0, n); coeftayl(%, z=0, n); end proc: seq(a(n), n=0..20); # _Vaclav Kotesovec_, Aug 03 2016

%t m = 12;

%t s = Series[1/(1 - 3 x - 5 y - 7 z + x y + 2 y z^2 + 3 x^2 z^2), {x, 0, m}, {y, 0, m}, {z, 0, m}] // Normal // Expand;

%t Join[{1}, Table[Coefficient[s, (x y z)^n], {n, 1, m}]] (* _Jean-François Alcover_, Jul 26 2016 *)

%o (PARI)

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

%o R = 1/(1-3*x-5*y-7*z+x*y+2*y*z^2+3*x^2*z^2);

%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])

%Y Cf. A268545-A268555.

%K nonn

%O 0,2

%A _Gheorghe Coserea_, Jul 19 2016