A274787 Diagonal of the rational function 1/(1-(wxyz + wxy + wxz + wy + xz + y + z)).

1, 7, 109, 2347, 59161, 1630447, 47569789, 1443991003, 45131425129, 1442569728607, 46937386025509, 1549447085182003, 51764874058957801, 1746923396870038303, 59463803615708931709, 2039212029425398630747, 70386791601625688812009, 2443437949352620443258463, 85253533514289277890811669

Offset

0,2

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..630

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+42*x^3-143*x^4-372*x^5-143*x^6+42*x^7-x^8)*y''' + (-3*x+195*x^2-1116*x^3-1674*x^4-171*x^5+183*x^6-6*x^7)*y'' + (-1+148*x-1565*x^2-560*x^3+325*x^4+124*x^5-7*x^6)*y' + (7-213*x+278*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)*(22*n^3 - 66*n^2 + 50*n - 13)*a(n-1) - (2*n - 3)*(230*n^4 - 1380*n^3 + 2790*n^2 - 2160*n + 541)*a(n-2) + (2*n - 5)*(2*n - 1)*(22*n^3 - 132*n^2 + 248*n - 137)*a(n-3) - (n-3)^3*(n-1)*(2*n - 1)*a(n-4).

a(n) ~ 10^(1/4) * (19 + 6*sqrt(10))^(n + 1/2) / (8 * Pi^(3/2) * n^(3/2)). (End)

Mathematica

a[n_] := SeriesCoefficient[1/(1 - (w x y z + w x y + w x z + w y + x z + y + z)), {w, 0, n}, {x, 0, n}, {y, 0, n}, {z, 0, n}];
Table[a[n], {n, 0, 18}] (* 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*y+w*x*z+w*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: A333092 A274668 A239848 * A116875 A357395 A371317

Adjacent sequences: A274784 A274785 A274786 * A274788 A274789 A274790

Keyword

nonn

Author

Gheorghe Coserea, Jul 14 2016

Status

approved