OFFSET
0,2
COMMENTS
Annihilating differential operator: x*(5*x+4)*(x^3+19*x^2+27*x-1)*Dx^2 + (15*x^4+206*x^3+363*x^2+216*x-4)*Dx + 5*x^3+33*x^2+36*x+28.
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.
FORMULA
G.f.: hypergeom([1/12, 5/12],[1],13824*x^4*(1-27*x-19*x^2-x^3)/(1-28*x+54*x^2+20*x^3+x^4)^3)/(1-28*x+54*x^2+20*x^3+x^4)^(1/4).
0 = x*(5*x+4)*(x^3+19*x^2+27*x-1)*y'' + (15*x^4+206*x^3+363*x^2+216*x-4)*y' + (5*x^3+33*x^2+36*x+28)*y, where y is the g.f.
From Peter Bala, Jan 15 2020: (Start)
a(n) = Sum_{0 <= j, k <= n} (-1)^(n+k)*C(n,k)*C(n,j)*C(n+k,k)*C(n+k+j,k+j). Cf. A001850 and A126086.
n^2*(37*n - 49)*a(n) = (999*n^3 - 2322*n^2 + 1567*n - 328)*a(n-1) + (703*n^3 - 2337*n^2 + 2295*n - 536)*a(n-2) + (n - 2)^2*(37*n - 12)*a(n-3). (End)
MATHEMATICA
gf = Hypergeometric2F1[1/12, 5/12, 1, 13824*x^4*(1 - 27*x - 19*x^2 - x^3) / (1 - 28*x + 54*x^2 + 20*x^3 + x^4)^3]/(1 - 28*x + 54*x^2 + 20*x^3 + x^4)^(1/4);
CoefficientList[gf + O[x]^20, x] (* Jean-François Alcover, Dec 01 2017 *)
PROG
(PARI)
my(x='x, y='y, z='z);
R = 1/(1 - x - y - z - x*y + x*z - 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 = 21; x = 'x + O('x^N);
Vec(hypergeom([1/12, 5/12], [1], 13824*x^4*(1-27*x-19*x^2-x^3)/(1-28*x+54*x^2+20*x^3+x^4)^3, N)/(1-28*x+54*x^2+20*x^3+x^4)^(1/4))
CROSSREFS
KEYWORD
nonn
AUTHOR
Gheorghe Coserea, Jul 05 2016
STATUS
approved