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A274673 Diagonal of the rational function 1/(1 - x - y - z - x y + x y z). 1
1, 7, 127, 2851, 70651, 1853377, 50452459, 1409575699, 40147379587, 1160568048157, 33947097696337, 1002532535965429, 29843356238833879, 894349641410968477, 26955867982764111427, 816484373069154316051, 24838062486275592671587, 758470289246834941140037, 23239359305672548933204261 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Annihilating differential operator: x*(5*x^2+4*x-6)*(x^4-3*x^3-27*x^2-64*x+2)*Dx^2 + (15*x^6-14*x^5-201*x^4-144*x^3+220*x^2+768*x-12)*Dx + 5*x^5+3*x^4+136*x^2+132*x+84.

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^3*(2-64*x-27*x^2-3*x^3+x^4)/(1-28*x-18*x^2-4*x^3+x^4)^3)/(1-28*x-18*x^2-4*x^3+x^4)^(1/4).

0 = x*(5*x^2+4*x-6)*(x^4-3*x^3-27*x^2-64*x+2)*y'' + (15*x^6-14*x^5-201*x^4-144*x^3+220*x^2+768*x-12)*y' + (5*x^5+3*x^4+136*x^2+132*x+84)*y, where y is the g.f.

Recurrence: 2*n^2*(517*n^2 - 1887*n + 1598)*a(n) = 2*(16544*n^4 - 76928*n^3 + 115285*n^2 - 64789*n + 11484)*a(n-1) + (13959*n^4 - 78867*n^3 + 152654*n^2 - 115150*n + 24984)*a(n-2) + (1551*n^4 - 10314*n^3 + 22982*n^2 - 18819*n + 3996)*a(n-3) - (n-3)^2*(517*n^2 - 853*n + 228)*a(n-4). - Vaclav Kotesovec, Jul 07 2016

MATHEMATICA

gf = Hypergeometric2F1[1/12, 5/12, 1, 1728*x^3*(2 - 64*x - 27*x^2 - 3*x^3 + x^4)/(1 - 28*x - 18*x^2 - 4*x^3 + x^4)^3]/(1 - 28*x - 18*x^2 - 4*x^3 + x^4)^(1/4);

CoefficientList[gf + O[x]^20, x] (* Jean-Fran├žois Alcover, Dec 01 2017 *)

PROG

(PARI)

my(x='x, y='y, z='z);

R = 1/(1 - x - y - z - x*y + x*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(10, R, [x, y, z])

(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^3*(2-64*x-27*x^2-3*x^3+x^4)/(1-28*x-18*x^2-4*x^3+x^4)^3, N)/(1-28*x-18*x^2-4*x^3+x^4)^(1/4))

CROSSREFS

Cf. A268545-A268555.

Sequence in context: A025166 A241955 A139291 * A215066 A092676 A002067

Adjacent sequences:  A274670 A274671 A274672 * A274674 A274675 A274676

KEYWORD

nonn

AUTHOR

Gheorghe Coserea, Jul 06 2016

STATUS

approved

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Last modified April 23 22:17 EDT 2019. Contains 322388 sequences. (Running on oeis4.)