A268555 Diagonal of the rational function of six variables 1/((1 - w - u v - u v w) * (1 - z - x y)).

1, 6, 78, 1260, 22470, 424116, 8305836, 166929048, 3419932230, 71109813060, 1496053026468, 31777397077608, 680354749147164, 14664155597771400, 317877850826299800, 6924815555276838960, 151505459922479997510, 3327336781596164286180

Offset: 0

Author

N. J. A. Sloane, Feb 29 2016

Keywords

• nonn
• easy

Comments

Also diagonal of rational function R(x,y,z) = 1 /(1 - x - y - z - x*y + x*z).

Annihilating differential operator: x*(16*x^2-24*x+1)*(d/dx)^2 + (48*x^2-48*x+1)*(d/dx) + 12*x-6.

Examples

			G.f. = 1 + 6*x + 78*x^2 + 1260*x^3 + 22470*x^4 + 424116*x^5 + 8305836*x^6 + ...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..200
• 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"

Crossrefs

Cf. A268545-A268555. Cf. A000984, A001850.

Programs

GAP

List([0..20],n->Binomial(2*n,n)*Sum([0..n],k->Binomial(n,k)*Binomial(n+k,k))); # Muniru A Asiru, Mar 19 2018

Maple

A268555 := proc(n)
    1/(1-w-u*v-u*v*w)/(1-z-x*y) ;
    coeftayl(%,x=0,n) ;
    coeftayl(%,y=0,n) ;
    coeftayl(%,z=0,n) ;
    coeftayl(%,u=0,n) ;
    coeftayl(%,v=0,n) ;
    coeftayl(%,w=0,n) ;
end proc:
seq(A268555(n),n=0..40) ; # R. J. Mathar, Mar 10 2016
seq(binomial(2*n,n)*add(binomial(n,k)*binomial(n+k,k), k = 0..n), n = 0..20); # Peter Bala, Mar 19 2018

Mathematica

sc = SeriesCoefficient;
a[n_] := 1/(1-w-u*v-u*v*w)/(1-z-x*y) // sc[#, {x, 0, n}]& // sc[#, {y, 0, n}]& // sc[#, {z, 0, n}]& // sc[#, {u, 0, n}]& // sc[#, {v, 0, n}]& // sc[#, {w, 0, n}]&;
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Nov 14 2017 *)
a[n_] := Product[Hypergeometric2F1[-n, -n, 1, i], {i, 1, 2}];
Table[a[n], {n, 0, 17}]  (* Peter Luschny, Mar 19 2018 *)

PARI

\\ system("wget http://www.jjj.de/pari/hypergeom.gpi");
read("hypergeom.gpi");
N = 18; x = 'x + O('x^N);
Vec(hypergeom_sym([1/12, 5/12],[1],6912*x^4*(1-24*x+16*x^2)/(1-24*x+48*x^2)^3, N)/(1-24*x+48*x^2)^(1/4)) \\ Gheorghe Coserea, Jul 05 2016

PARI

{a(n) = if( n<1, n==0, my(A = vector(n+1)); A[1] = 1; A[2] = 6; for(k=2, n, A[k+1] = (6*(2*k-1)^2*A[k] - 4*(2*k-1)*(2*k-3)*A[k-1]) / k^2); A[n+1])}; /* Michael Somos, Jan 22 2017 */

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), 18)
\\ test: diag(1/(1-x-y-z-x*y+x*z)) == diag(1/((1-w-u*v-u*v*w)*(1-z-x*y)))
\\ Gheorghe Coserea, Jun 15 2018

Formula

D-finite with recurrence: n^2*a(n) -6*(2*n-1)^2*a(n-1) +4*(2*n-1)*(2*n-3)*a(n-2)=0. - R. J. Mathar, Mar 10 2016

a(n) ~ sqrt(4+3*sqrt(2)) * 2^(2*n-3/2) * (1+sqrt(2))^(2*n) / (Pi*n). - Vaclav Kotesovec, Jul 01 2016

G.f.: hypergeom([1/12, 5/12],[1],6912*x^4*(1-24*x+16*x^2)/(1-24*x+48*x^2)^3)/(1-24*x+48*x^2)^(1/4).

0 = x*(16*x^2-24*x+1)*y'' + (48*x^2-48*x+1)*y' + (12*x-6)*y, where y is g.f.

a(n) = A000984(n)*A001850(n) = C(2*n,n)*Sum_{k = 0..n} C(n,k)*C(n+k,k). - Peter Bala, Mar 19 2018

a(n) = [(x*y)^n] (1 + x + y + 2*x*y)^(2*n), so the sequence is the diagonal of the three variable rational function 1/(1 - u*(1 + x + y + 2*x*y)^2). - Peter Bala, Oct 30 2024