OFFSET
0,2
COMMENTS
Expand the rational function (1+x*y*z)/(1-x-y-z) as Sum_i Sum_j Sum_k c(i,j,k)*x^i*y^j*z^k; a(n) = c(n,n,n).
If the numerator is changed to 1, we get A006480.
Suggested by Christol's Conjecture (see reference).
REFERENCES
Abdelaziz, Youssef, C. Koutschan, and J. M. Maillard. "On Christol’s conjecture." Journal of Physics A: Mathematical and Theoretical 53.20 (2020): 205201; arXiv:1912.10259.
LINKS
Robert Israel, Table of n, a(n) for n = 0..300
Y. Abdelaziz, C. Koutschan, and J-M. Maillard, On Christol's conjecture, arXiv:1912.10259 [math.NT], 2019-2020.
FORMULA
Conjectures from Robert Israel, Oct 25 2020: (Start)
G.f.: (x + 1)*LegendreP(-1/3, 1 - 54*x).
(-27*n^2 - 27*n - 6)*a(n + 1) + (-53*n^2 - 214*n - 173)*a(n + 2) + (-25*n^2 - 179*n - 319)*a(n + 3) + (n^2 + 8*n + 16)*a(n + 4) = 0. (End)
a(n) = (28*n^2 - 27*n + 6) * (3*n)! / (3 * (3*n - 1) * (3*n - 2) * n!^3). - Vaclav Kotesovec, Oct 28 2020
MAPLE
N:= 25: # for a(0)..a(N)
F:= (1+x*y*z)/(1-x-y-z):
S1:= series(F, x, N+1):
L1:= [seq(coeff(S1, x, i), i=0..N)]:
L2:= [seq(coeff(series(L1[i+1], y, i+1), y, i), i=0..N)]:
seq(coeff(series(L2[i+1], z, i+1), z, i), i=0..N); # Robert Israel, Oct 25 2020
MATHEMATICA
nmax = 20; Flatten[{1, Table[Coefficient[Series[(1 + x*y*z)/(1 - x - y - z), {x, 0, n}, {y, 0, n}, {z, 0, n}], x^n*y^n*z^n], {n, 1, nmax}]}] (* Vaclav Kotesovec, Oct 23 2020 *)
PROG
(PARI) {a(n) = if(n==0, 1, (3*(n-1))!/(n-1)!^3+(3*n)!/n!^3)} \\ Seiichi Manyama, Oct 31 2020
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Oct 22 2020
EXTENSIONS
More terms from Vaclav Kotesovec, Oct 23 2020
STATUS
approved