A274786 Diagonal of the rational function 1/(1 - (wxz + wy + wz + xy + xz + y + z)).

1, 6, 114, 2940, 87570, 2835756, 96982116, 3446781624, 126047377170, 4712189770860, 179275447715364, 6918537571788024, 270178056420497316, 10656693484898995800, 423937118582497715400, 16989669600664370275440, 685277433339552643145490, 27797911234749454227812460, 1133299570662800455270517700

Offset

0,2

Links

Gheorghe Coserea, Table of n, a(n) for n = 0..200

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.

Timothy Huber, Daniel Schultz, and Dongxi Ye, Ramanujan-Sato series for 1/pi, Acta Arith. (2023) Vol. 207, 121-160. See p. 11.

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

Formula

a(n) = Sum_{j=0..2*n} (-1)^j * binomial(2*n,j) * binomial(j,n)^3.

a(n) = T(2*n,n), where triangle T(n,k) is defined by A262704.

0 = (-x^2+44*x^3+16*x^4)*y''' + (-3*x+198*x^2+96*x^3)*y'' + (-1+144*x+108*x^2)*y' + (6+12*x)*y, where y is the g.f.

Recurrence: n^3*a(n) = 2*(2*n - 1)*(11*n^2 - 11*n + 3)*a(n-1) + 4*(n-1)*(2*n - 3)*(2*n - 1)*a(n-2). - Vaclav Kotesovec, Dec 01 2017

a(n) ~ 2^(2*n - 1) * phi^(5*n + 5/2) / (5^(1/4) * (Pi*n)^(3/2)), where phi = A001622 = (1 + sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Dec 01 2017

Conjecture: a(n) = [x^n] (1 + x)^(2*n) * P(n,(1 + x)/(1 - x))^2, where P(n,x) denotes the n-th Legendre polynomial. Cf. A005260(n) = [x^n] (1 - x)^(2*n) * P(n,(1 + x)/(1 - x))^2, due to Carlitz. - Peter Bala, Sep 21 2021

a(n) = A000984(n) * A005258(n). - Peter Bala, Oct 12 2024

Mathematica

a[n_] := Sum[(-1)^j Binomial[2n, j] Binomial[j, n]^3, {j, n, 2n}];
(* or much faster *)
a[0] = 1; a[1] = 6; a[n_] := a[n] = (2*(2*n - 1)*(11*n^2 - 11*n + 3)*a[n - 1] + 4*(n - 1)*(2*n - 3)*(2*n - 1)*a[n - 2])/n^3;
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Dec 01 2017, after Vaclav Kotesovec *)

Prog

(PARI)
a(n) = sum(j=n, 2*n, (-1)^(j)*binomial(2*n, 2*n - j)*binomial(j, n)^3);
(PARI)
my(x='x, y='y, z='z, w='w);
R = 1/(1-(w*x*z+w*y+w*z+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(18, R, [x, y, z, w])

Crossrefs

Cf. A000984, A005258, A262704, A268545 - A268555.

Sequence in context: A194476 A059116 A121544 * A317172 A278752 A003425

Adjacent sequences: A274783 A274784 A274785 * A274787 A274788 A274789

Keyword

nonn

Author

Gheorghe Coserea, Jul 14 2016

Status

approved