A275048 Diagonal of the rational function 1/(1 - 3x - 5y - 7z + xy + 2yz^2 + 3x^2z^2).

1, 616, 947175, 1812651820, 3833011883965, 8582819380142616, 19946071353510410136, 47578122531207001944168, 115702070514540009854741415, 285583642613093627090885877280, 713269435359072253352128013072035, 1798640871397075848520426334656821360, 4571867984920021228226177292547960261132

Offset

0,2

Links

Gheorghe Coserea and Vaclav Kotesovec, Table of n, a(n) for n = 0..134 (terms 0..55 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.

Jacques-Arthur Weil, Supplementary Material for the Paper "Diagonals of rational functions and selected differential Galois groups"

S. Boukraa, S. Hassani, J-M. Maillard and J-A. Weil, Differential algebra on lattice Green functions and Calabi-Yau operators (unabridged  version), arXiv:1311.2470 [math-ph], 2013.

Formula

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

a(n) ~ c*d^n/n, where d = 2768.26905153125473648033608782909246512063674470098..., c = 0.275933667674847170980793689... . - Vaclav Kotesovec, Aug 03 2016

Maple

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

Mathematica

m = 12;
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;
Join[{1}, Table[Coefficient[s, (x y z)^n], {n, 1, m}]] (* Jean-François Alcover, Jul 26 2016 *)

Prog

(PARI)
my(x='x, y='y, z='z);
R = 1/(1-3*x-5*y-7*z+x*y+2*y*z^2+3*x^2*z^2);
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(12, R, [x, y, z])

Crossrefs

Cf. A268545-A268555.

Sequence in context: A332161 A221040 A221503 * A350083 A275739 A108818

Adjacent sequences: A275045 A275046 A275047 * A275049 A275050 A275051

Keyword

nonn

Author

Gheorghe Coserea, Jul 19 2016

Status

approved