A274670 Diagonal of the rational function 1/(1 - x - y - z - x y + x z - x y z).

1, 7, 103, 1891, 38371, 824377, 18379579, 420563731, 9810403267, 232264240957, 5564072675833, 134574852764821, 3280845731941519, 80522277272406613, 1987608338377888483, 49305191067563987731, 1228368016027453079587, 30719511029184435338053, 770839386237255136597501

Offset

0,2

Comments

Annihilating differential operator: x*(8*x^2-5*x-24) * (2*x^4-9*x^3+205*x^2-167*x+6)*Dx^2 + (48*x^6-184*x^5+1535*x^4-1186*x^3-13973*x^2+8016*x-144)*Dx + 16*x^5-36*x^4-491*x^3+839*x^2-3840*x+1008.

Also diagonal of rational functions 1/(1 + y + 3*z + x*y + y*z + x*z + x*y*z), 1/(1 - 2*y - z - x*y - y*z - x*z - x*y*z), 1/(1 - 3*x + y + z - 2*x*y + y*z - x*z - x*y*z), 1/(1 - 2*x - y - z - x*y + y*z + x*z + x*y*z), 1/(1 - x - y - z + x*y - 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"

Formula

G.f.: hypergeom([1/12, 5/12],[1],1728*x^4*(6-167*x+205*x^2-9*x^3+2*x^4)/(1-28*x+78*x^2-4*x^3+x^4)^3)/(1-28*x+78*x^2-4*x^3+x^4)^(1/4).

0 = x*(8*x^2-5*x-24)*(2*x^4-9*x^3+205*x^2-167*x+6)*y'' + (48*x^6-184*x^5+1535*x^4-1186*x^3-13973*x^2+8016*x-144)*y' + (16*x^5-36*x^4-491*x^3+839*x^2-3840*x+1008)*y, where y is the g.f.

D-finite with recurrence 144*(n^2)*a(n) +6*(-663*n^2+653*n-158)*a(n-1) +(4037*n^2-6212*n+116)*a(n-2) +(2145*n^2-13829*n+21343)*a(n-3) +(-1637*n^2+13198*n-26109)*a(n-4) +2*(41*n^2-359*n+788)*a(n-5) -16*(n-5)^2*a(n-6)=0. - R. J. Mathar, Jul 27 2022

Mathematica

gf = Hypergeometric2F1[1/12, 5/12, 1, 1728*x^4*(6 - 167*x + 205*x^2 - 9*x^3 + 2*x^4)/(1 - 28*x + 78*x^2 - 4*x^3 + x^4)^3]/(1 - 28*x + 78*x^2 - 4*x^3 + x^4)^(1/4);
CoefficientList[gf + O[x]^20, x] (* Jean-François Alcover, Dec 01 2017 *)

Prog

(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], 1728*x^4*(6-167*x+205*x^2-9*x^3+2*x^4)/(1-28*x+78*x^2-4*x^3+x^4)^3, N)/(1-28*x+78*x^2-4*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 - x*y*z), 19)
\\ test: diag(1/(1 - x - y - z - x*y + x*z - x*y*z)) == diag(1/(1 - 2*x - y - z - x*y + y*z + x*z + x*y*z))
\\ Gheorghe Coserea, Jul 03 2018

Crossrefs

Cf. A268545-A268555.

Sequence in context: A032460 A267234 A246914 * A188946 A368440 A365014

Adjacent sequences: A274667 A274668 A274669 * A274671 A274672 A274673

Keyword

nonn

Author

Gheorghe Coserea, Jul 05 2016

Status

approved