A275047 Diagonal of the rational function 1/(1-(1+w)(xy + xz + yz)) [even-indexed terms only].

1, 18, 1350, 141120, 17151750, 2272538268, 318430816704, 46404203788800, 6961609406993670, 1068002895589987500, 166779781860762170100, 26422986893371642828800, 4236593267629481817240000, 686167053247777413372681600, 112093956900827388909570240000

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-(1+w)*(x*y + x*z + y*z)).

From Peter Bala, Jun 22 2023: (Start)

a(n) = A(n,n,2*n,2*n) (= A(2*n,2*n,n,n)) in the notation of Straub, equation 8, where it is shown that the supercongruences a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) hold for all primes p >= 5 and all positive integers n and r. This also follows from Meštrović equation 39, since a(n) = binomial(3*n,n)^2 * binomial(2*n,n).

Inductively define a family of sequences {a(i,n) : n >= 0}, i >= 1, by setting a(1,n) = a(n) and, for i >= 2, a(i,n) = [x^n] ( exp(Sum_{k >= 1} a(i-1,k)*x^k/k) )^n.

We conjecture that the sequences {a(i,n) : n >= 0}, i >= 2, also satisfy the supercongruences u(n*p^r) == u(n*p^(r-1)) (mod p^(3*r)) for all primes p >= 5, and positive integers n and r. Cf. A362725 and A362732. (End)

Links

Alois P. Heinz, Table of n, a(n) for n = 0..444 (first 34 terms from Gheorghe Coserea)

A. Bostan, S. Boukraa, J.-M. Maillard and J.-A. Weil, Diagonals of rational functions and selected differential Galois groups, arXiv preprint arXiv:1507.03227 [math-ph], 2015.

R. Meštrović, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2012), arXiv:1111.3057 [math.NT], 2011.

Armin Straub, Multivariate Apéry numbers and supercongruences of rational functions, Algebra & Number Theory, Vol. 8, No. 8 (2014), pp. 1985-2008; arXiv preprint, arXiv:1401.0854 [math.NT], 2014.

Jacques-Arthur Weil, Supplementary Material for the Paper "Diagonals of rational functions and selected differential Galois groups"

Formula

0 = (-4*x^2+729*x^4)*y'''' + (-20*x+7290*x^3)*y''' + (-16+18063*x^2)*y'' + 10449*x*y' + 576*y, where y = 1 + 18*x^2 + 1350*x^4 + ...

From Vaclav Kotesovec, Aug 03 2016: (Start)

a(n) = (3*n)!^2 / (n!^4 * (2*n)!).

a(n) ~ 3^(6*n+1) / (Pi^(3/2) * n^(3/2) * 2^(2*n+2)).

(End)

G.f.: 4F3(1/3,1/3,2/3,2/3;1/2,1,1;729x/4). - Benedict W. J. Irwin, Aug 05 2016

From Peter Bala, Sep 20 2021: (Start)

a(n) = 9*(3*n - 1)^2*(3*n - 2)^2/(2*n^3*(2*n - 1))*a(n-1).

a(n) = Sum_{k = n..3*n} (-1)^k*binomial(3*n,k)^2*binomial(k,n)^2. (End)

From Peter Bala, Jun 22 2023: (Start)

a(n) = binomial(3*n,n)^2 * binomial(2*n,n) = A188662(n) * A000984(n).

a(n) = Sum_{k = 0..n} binomial(n,k)*binomial(2*n,k)*binomial(2*n-k,n)* binomial(4*n-k,2*n).

a(n) = [(x*y)^n * (z*t)^(2*n)] 1/((1 - x - y)*(1 - z - t) - x*y*z*t). (End)

Example

1 + 18*x^2 + 1350*x^4 + 141120*x^6 + ...

Maple

a:= proc(n) option remember; `if`(n=0, 1,
      9*(3*n-1)^2*(3*n-2)^2*a(n-1)/((4*n-2)*n^3))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 25 2016

Mathematica

Table[(3*n)!^2 / (n!^4*(2*n)!), {n, 0, 20}] (* Vaclav Kotesovec, Aug 03 2016 *)
CoefficientList[Series[HypergeometricPFQ[{1/3, 1/3, 2/3, 2/3}, {1/2, 1, 1}, 729x/4], {x, 0, 10}], x] (* Benedict W. J. Irwin, Aug 05 2016 *)

Prog

(PARI)
my(x='x, y='y, z='z, w='w);
R = 1/(1-(1+w)*(x*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(23, R, [x, y, z, w])

Crossrefs

Cf. A000984, A188662, A268545 - A268555.

Sequence in context: A252969 A182286 A292609 * A160252 A276015 A210823

Adjacent sequences: A275044 A275045 A275046 * A275048 A275049 A275050

Keyword

nonn,easy

Author

Gheorghe Coserea, Jul 18 2016

Status

approved