A274671 Diagonal of the rational function 1/(1 - x - y - z - x y + x z + y z - x y z).

1, 5, 49, 593, 7921, 111965, 1641865, 24705665, 378931585, 5898329045, 92893693729, 1477015762865, 23671209428881, 381902943661517, 6196712221450009, 101043805539177473, 1654726991239056385, 27201210101330189477, 448652354194417534609, 7422254208909904273553

Offset: 0

Author

Gheorghe Coserea, Jul 05 2016

Keywords

• nonn

Comments

Annihilating differential operator: x*(4*x-5)*(x^3+27*x^2-19*x+1)*Dx^2 + (12*x^4+196*x^3-481*x^2+190*x-5)*Dx + 4*x^3+18*x^2-95*x+25.

Also diagonal of rational functions 1/(1 + z + 2*x*y - y*z + x*z - x*y*z), 1/(1 + 2*z + x*y - 2*y*z + x*z - x*y*z), 1/(1 + y + z + 2*x*y + y*z + x*z + x*y*z), 1/(1 + y + 2*z + x*y + 2*y*z + x*z + x*y*z), 1/(1 + x + y - 2*z - y*z - 2*x*z + x*y*z). - Gheorghe Coserea, Jul 03 2018

Links

• Gheorghe Coserea, Table of n, a(n) for n = 0..310
• 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"

Crossrefs

Cf. A268545-A268555.

Programs

Mathematica

gf = Hypergeometric2F1[1/12, 5/12, 1, 13824*x^5*(1 - 19*x + 27*x^2 + x^3) / (1 - 20*x + 54*x^2 + 28*x^3 + x^4)^3]/(1 - 20*x + 54*x^2 + 28*x^3 + x^4)^(1/4);
CoefficientList[gf + O[x]^20, x] (* Jean-François Alcover, Dec 01 2017 *)
a[n_] := HypergeometricPFQ[{-n, -n, n + 1}, {1, 1}, 2];
Table[a[n], {n, 0, 19}]  (* Peter Luschny, Nov 15 2021 *)

PARI

\\ system("wget http://www.jjj.de/pari/hypergeom.gpi");
read("hypergeom.gpi");
N = 20; x = 'x + O('x^N);
Vec(hypergeom([1/12, 5/12],[1],13824*x^5*(1-19*x+27*x^2+x^3)/(1-20*x+54*x^2+28*x^3+x^4)^3, N)/(1-20*x+54*x^2+28*x^3+x^4)^(1/4))

PARI

diag(expr, N=22, var=variables(expr)) = {
  my(a = vector(N));
  for (k = 1, #var, expr = taylor(expr, var[#var - k + 1], N));
  for (n = 1, N, a[n] = expr;
    for (k = 1, #var, a[n] = polcoeff(a[n], n-1)));
  return(a);
};
diag(1/(1 - x - y - z - x*y + x*z + y*z - x*y*z), 20)
\\ test: diag(1/(1 - x - y - z - x*y + x*z + y*z - x*y*z)) == diag(1/(1 + x + y - 2*z - y*z - 2*x*z + x*y*z))
\\ Gheorghe Coserea, Jul 03 2018

Formula

G.f.: hypergeom([1/12, 5/12],[1],13824*x^5*(1-19*x+27*x^2+x^3)/(1-20*x+54*x^2+28*x^3+x^4)^3)/(1-20*x+54*x^2+28*x^3+x^4)^(1/4).

0 = x*(4*x-5)*(x^3+27*x^2-19*x+1)*y'' + (12*x^4+196*x^3-481*x^2+190*x-5)*y' + (4*x^3+18*x^2-95*x+25)*y, where y is the g.f.

a(n) = Sum_{k=0..n} binomial(n,k)^2 * binomial(n+k,k) * 2^k. - Ilya Gutkovskiy, Nov 15 2021

a(n) = hypergeom([-n, -n, n + 1], [1, 1], 2). - Peter Luschny, Nov 15 2021

Recurrence: n^2*(37*n - 62)*a(n) = (703*n^3 - 1881*n^2 + 1383*n - 330)*a(n-1) - (999*n^3 - 3672*n^2 + 4267*n - 1510)*a(n-2) - (n-2)^2*(37*n - 25)*a(n-3). - Vaclav Kotesovec, Nov 15 2021