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A275050 Diagonal of the rational function 1/(1-(wxz + wyz + xyz + wy + xy + z)) [even-indexed terms only]. 1
1, 36, 5580, 1209600, 305127900, 83936348496, 24422566424256, 7391145688692480, 2302861234904415900, 733755111903173646000, 237987702318837667276080, 78313025454309175928186880, 26080521003090619899885979200, 8773677817145303533293886560000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Odd-order terms are zero since R(x,y,z,w) = R(-x,-y,z,-w), where R(x,y,z,w) = 1/(1-(w*x*z+w*y*z+x*y*z+w*y+x*y+z)).

LINKS

Gheorghe Coserea and Alois P. Heinz, Table of n, a(n) for n = 0..389 (first 34 terms from Gheorghe Coserea)

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

a(n) = [(xyzw)^(2n)] 1/(1-(w*x*z+w*y*z+x*y*z+w*y+x*y+z)).

0 = (-50*x^3+19341*x^5-155898*x^7-95256*x^9)*y'''' + (-200*x^2+170982*x^4-1015686*x^6-1143072*x^8)*y''' + (-50*x+353295*x^3-1068420*x^5-3513888*x^7)*y'' + (50+141777*x^2+282540*x^4-2709504*x^6)*y' + (66816*x^3-254016*x^5)*y, where y(x) = A(x^2).

a(n) ~ sqrt(1 + 2*sqrt(3/11)) * (189 + 33*sqrt(33))^n / (2*Pi^(3/2)*n^(3/2)). - Vaclav Kotesovec, Aug 03 2016

EXAMPLE

1 + 36*x^2 + 5580*x^4 + 1209600*x^6 + ...

MAPLE

a:= proc(n) option remember; `if`(n<2, 36^n, (9*(n-1)*

       (2*n-1)*(693*n^3-1554*n^2+989*n-160)*a(n-1)+

       (6*(33*n-8))*(3*n-4)*(2*n-1)*(2*n-3)*(3*n-5)*

        a(n-2)) / ((n-1)*(33*n-41)*n^3))

    end:

seq(a(n), n=0..20);  # Alois P. Heinz, Jul 26 2016

MATHEMATICA

a[n_] := a[n] = If[n < 2, 36^n, (9*(n - 1)*(2*n - 1)*(693*n^3 - 1554*n^2 + 989*n - 160)*a[n - 1] + (6*(33*n - 8))*(3*n - 4)*(2*n - 1)*(2*n - 3)*(3*n - 5)*a[n - 2])/((n - 1)*(33*n - 41)*n^3)];

Table[a[n], {n, 0, 20}] (* Jean-Fran├žois Alcover, Sep 07 2017, after Alois P. Heinz *)

PROG

(PARI)

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

R = 1/(1-(w*x*z+w*y*z+x*y*z+w*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(22, R, [x, y, z, w])

CROSSREFS

Cf. A268545-A268555.

Sequence in context: A299698 A203752 A184135 * A222336 A291975 A307351

Adjacent sequences:  A275047 A275048 A275049 * A275051 A275052 A275053

KEYWORD

nonn

AUTHOR

Gheorghe Coserea, Jul 19 2016

STATUS

approved

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Last modified September 27 17:05 EDT 2021. Contains 347693 sequences. (Running on oeis4.)