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A268545 From the diagonal of 1/(1 - (y + z + x w + x z w + x y w)). 70
1, 10, 246, 7540, 255430, 9163980, 341237820, 13042646760, 508236930630, 20101587623260, 804500381097556, 32508382071448920, 1324112273705453596, 54296281503438398200, 2239266766596344681400, 92809720054802928741840, 3863305447624183692730950, 161427619265399264526790140 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

From Gheorghe Coserea, Jul 03 2016: (Start)

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

Annihilating differential operator: x*(4*x+3)*(16*x^2+44*x-1)*Dx^2 + (128*x^3+320*x^2+264*x-3)*Dx + 16*x^2+12*x+30.

(End)

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.

Steffen Eger, On the Number of Many-to-Many Alignments of N Sequences, arXiv:1511.00622 [math.CO], 2015.

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

FORMULA

Conjecture: n^2*(10*n-13)*a(n) +2*(-220*n^3+506*n^2-334*n+63)*a(n-1) -4*(10*n-3)*(-3+2*n)^2*a(n-2)=0. - R. J. Mathar, Apr 15 2016

a(n) ~ (1+sqrt(5))^(5*n+2) / (5^(1/4) * Pi * n * 2^(3*n+3)). - Vaclav Kotesovec, Jul 01 2016

G.f.: hypergeom([1/12, 5/12],[1],6912*x^3*(1-44*x-16*x^2)/(1-40*x+16*x^2)^3)/(1-40*x+16*x^2)^(1/4). - Gheorghe Coserea, Jul 01 2016

0 = x*(4*x+3)*(16*x^2+44*x-1)*y'' + (128*x^3+320*x^2+264*x-3)*y' + (16*x^2+12*x+30)*y, where y is the g.f. - Gheorghe Coserea, Jul 03 2016

a(n) = Sum_{j = 0..n} C(n,j)*Sum_{k = 0..j} C(j,k)*C(n+k,j)*C(2*n+k,n) (apply Eger, Theorem 3 to the set of column vectors S = {[1,0,0], [0,1,0], [0,0,1], [1,1,0], [0,1,1]}. - Peter Bala, Jan 26 2018

MAPLE

A268545 := proc(n)

    1/(1-y-z-x*w-x*z*w-x*y*w) ;

    coeftayl(%, x=0, n) ;

    coeftayl(%, y=0, n) ;

    coeftayl(%, z=0, n) ;

    coeftayl(%, w=0, n) ;

end proc:

seq(A268545(n), n=0..40) ; # R. J. Mathar, Apr 15 2016

#alternative program

with(combinat): seq(add(binomial(n, j)*add(binomial(j, k)*binomial(n+k, j)*binomial(2*n+k, n), k = 0..j), j = 0..n), n = 0..20); # Peter Bala, Jan 26 2018

MATHEMATICA

a[n_] := a[n] = 1/(1 - y - z - x*w - x*z*w - x*y*w) // SeriesCoefficient[#, {x, 0, n}]& // SeriesCoefficient[#, {y, 0, n}]& // SeriesCoefficient[#, {z, 0, n}]& // SeriesCoefficient[#, {w, 0, n}]&;

Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 0, 40}]

(* or: *)

HypergeometricPFQ[{1/12, 5/12}, {1}, (6912*x^3*(-16*x^2 - 44*x + 1))/ (16*x^2 - 40*x + 1)^3]/(16*x^2 - 40*x + 1)^(1/4) + O[x]^41 // CoefficientList[#, x]& (* Jean-Fran├žois Alcover, Nov 12 2017, after Gheorghe Coserea *)

PROG

(PARI)

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

R =  1/(1 - x - y - z - x*y - 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(10, R, [x, y, z])

(PARI) \\ system("wget http://www.jjj.de/pari/hypergeom.gpi");

read("hypergeom.gpi");

N = 21; x = 'x + O('x^N);

Vec(hypergeom([1/12, 5/12], [1], 6912*x^3*(1-44*x-16*x^2)/(1-40*x+16*x^2)^3, N)/(1-40*x+16*x^2)^(1/4)) \\ Gheorghe Coserea, Jul 03 2016

CROSSREFS

Cf. A268545 - A268555.

Sequence in context: A249564 A034222 A197437 * A211093 A108792 A265854

Adjacent sequences:  A268542 A268543 A268544 * A268546 A268547 A268548

KEYWORD

nonn,easy,changed

AUTHOR

N. J. A. Sloane, Feb 29 2016

STATUS

approved

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Last modified February 21 04:22 EST 2018. Contains 299389 sequences. (Running on oeis4.)