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A268555 Diagonal of the rational function of six variables 1/((1 - w - u v - u v w) * (1 - z - x y)). 72
1, 6, 78, 1260, 22470, 424116, 8305836, 166929048, 3419932230, 71109813060, 1496053026468, 31777397077608, 680354749147164, 14664155597771400, 317877850826299800, 6924815555276838960, 151505459922479997510, 3327336781596164286180 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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.

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"

FORMULA

Conjecture: 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

EXAMPLE

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

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-Fran├žois 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 *)

PROG

(PARI)

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

R = 1 /(1 - x - y - z - x*y + x*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 = 18; x = 'x + O('x^N);

Vec(hypergeom([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 */

(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

CROSSREFS

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

Sequence in context: A074112 A197103 A208473 * A131926 A132866 A279444

Adjacent sequences:  A268552 A268553 A268554 * A268556 A268557 A268558

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Feb 29 2016

STATUS

approved

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Last modified May 27 00:38 EDT 2018. Contains 304689 sequences. (Running on oeis4.)