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A274780 Diagonal of the rational function 1/(1 - x - y - z - x y - y z - x y z). 1
1, 11, 283, 9155, 327811, 12436541, 489807991, 19803209843, 816330309475, 34156900690841, 1446223566321733, 61826502242685653, 2664286789334520559, 115586782462237980905, 5043474229642670729743, 221159937117980575239635, 9740104064284459778657635, 430601976748346102416423025 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Annihilating differential operator: x*(x-2)*(5*x+2)*(3*x^2-47*x+1)*Dx^2 + (45*x^4-506*x^3+157*x^2+380*x-4)*Dx + 15*x^3-63*x^2-136*x+44.

LINKS

Gheorghe Coserea, Table of n, a(n) for n = 0..310

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

G.f.: hypergeom([1/12, 5/12], [1],1728*(3*x^2-47*x+1)*(x-2)^2*x^3/(x^4-20*x^3+78*x^2-44*x+1)^3)/(x^4-20*x^3+78*x^2-44*x+1)^(1/4).

0 = x*(x-2)*(5*x+2)*(3*x^2-47*x+1)*y'' + (45*x^4-506*x^3+157*x^2+380*x-4)*y' + (15*x^3-63*x^2-136*x+44)*y, where y is the g.f.

Recurrence: 2*n^2*(39*n - 53)*a(n) = (3705*n^3 - 8740*n^2 + 5823*n - 1096)*a(n-1) - (2067*n^3 - 6943*n^2 + 6915*n - 1664)*a(n-2) + 3*(n-2)^2*(39*n - 14)*a(n-3). - Vaclav Kotesovec, Jul 07 2016

a(n) ~ sqrt(29/2 + 103/(2*sqrt(13))) * ((47+13*sqrt(13))/2)^n / (6*Pi*n). - Vaclav Kotesovec, Jul 07 2016

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

MAPLE

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

MATHEMATICA

gf = Hypergeometric2F1[1/12, 5/12, 1, 1728*(3*x^2 - 47*x + 1)*(x - 2)^2*x^3 / (x^4 - 20*x^3 + 78*x^2 - 44*x + 1)^3]/(x^4 - 20*x^3 + 78*x^2 - 44*x + 1)^(1/4);

CoefficientList[gf + O[x]^20, x] (* Jean-Fran├žois Alcover, Dec 01 2017 *)

PROG

(PARI)

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

R = 1/(1 - x - y - z - x*y - y*z - x*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 = 20; x = 'x + O('x^N);

Vec(hypergeom([1/12, 5/12], [1], 1728*(3*x^2-47*x+1)*(x-2)^2*x^3/(x^4-20*x^3+78*x^2-44*x+1)^3, N)/(x^4-20*x^3+78*x^2-44*x+1)^(1/4))

CROSSREFS

Cf. A268545-A268555.

Sequence in context: A129754 A103547 A171195 * A323311 A280359 A196790

Adjacent sequences:  A274777 A274778 A274779 * A274781 A274782 A274783

KEYWORD

nonn,easy

AUTHOR

Gheorghe Coserea, Jul 06 2016

STATUS

approved

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Last modified July 25 12:56 EDT 2021. Contains 346290 sequences. (Running on oeis4.)