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A274670 Diagonal of the rational function 1/(1 - x - y - z - x y + x z - x y z). 1
1, 7, 103, 1891, 38371, 824377, 18379579, 420563731, 9810403267, 232264240957, 5564072675833, 134574852764821, 3280845731941519, 80522277272406613, 1987608338377888483, 49305191067563987731, 1228368016027453079587, 30719511029184435338053, 770839386237255136597501 (list; graph; refs; listen; history; text; internal format)
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.

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-Fran├žois 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 A234292 A177752

Adjacent sequences:  A274667 A274668 A274669 * A274671 A274672 A274673

KEYWORD

nonn

AUTHOR

Gheorghe Coserea, Jul 05 2016

STATUS

approved

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Last modified July 8 08:08 EDT 2020. Contains 335520 sequences. (Running on oeis4.)