A274788 Diagonal of the rational function 1/(1-(wxyz + wxz + wy + wz + xy + xz + y + z)).

1, 7, 133, 3547, 109921, 3710287, 132371149, 4909790011, 187430229889, 7315689889207, 290621404873933, 11711948497012771, 477636896775866569, 19675331299610850871, 817461706854954936733, 34215970307619080633947, 1441443460101276484035169, 61071445002917964407145031

Offset

0,2

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..600

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"

Formula

0 = (x^2-46*x^3-17*x^4+60*x^5-17*x^6-46*x^7+x^8)*y''' + (3*x-213*x^2+180*x^3+270*x^4-333*x^5-201*x^6+6*x^7)*y'' + (1-160*x+545*x^2-160*x^3-481*x^4-136*x^5+7*x^6)*y' + (-7+105*x-170*x^2-22*x^3-3*x^4+x^5)*y, where y is g.f.

From Vaclav Kotesovec, Mar 19 2023: (Start)

Recurrence: (n-2)*n^3*(2*n - 5)*a(n) = (2*n - 5)*(2*n - 1)*(24*n^3 - 72*n^2 + 54*n - 13)*a(n-1) - (2*n - 3)*(78*n^4 - 468*n^3 + 928*n^2 - 678*n + 161)*a(n-2) + (2*n - 5)*(2*n - 1)*(24*n^3 - 144*n^2 + 270*n - 149)*a(n-3) - (n-3)^3*(n-1)*(2*n - 1)*a(n-4).

a(n) ~ sqrt(5 + 13/sqrt(5) + sqrt(579/10 + (53*sqrt(5))/2)) * (12 + 5*sqrt(5) + 2*sqrt(67 + 30*sqrt(5)))^n / (4 * Pi^(3/2) * n^(3/2)). (End)

Mathematica

a[n_] := SeriesCoefficient[1/(1 - (w x y z + w x z + w y + w z + x y + x z + y + z)), {w, 0, n}, {x, 0, n}, {y, 0, n}, {z, 0, n}];
Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Nov 16 2018 *)

Prog

(PARI)
my(x='x, y='y, z='z, w='w);
R = 1/(1-(w*x*y*z+w*x*z+w*y+w*z+x*y+x*z+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(12, R, [x, y, z, w])

Crossrefs

Cf. A268545-A268555.

Sequence in context: A110111 A245318 A370782 * A274258 A251577 A082164

Adjacent sequences: A274785 A274786 A274787 * A274789 A274790 A274791

Keyword

nonn

Author

Gheorghe Coserea, Jul 14 2016

Status

approved