A338075 Diagonal terms in the expansion of (1+x*y*z)/(1-x-y-z).

1, 7, 96, 1770, 36330, 791406, 17909892, 416226096, 9864584730, 237338943270, 5778870222840, 142077992254380, 3521258757984240, 87862829835387600, 2205050763983594400, 55615552451285359680, 1408840444191389714010, 35825204161237194511830, 914089586182634239686000

Offset: 0

Author

N. J. A. Sloane, Oct 22 2020

Keywords

• nonn

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.

Crossrefs

Other examples arising from diagonal terms of multivariate g.f.s: A000172, A006480, A338076.

Programs

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 *)

PARI

{a(n) = if(n==0, 1, (3*(n-1))!/(n-1)!^3+(3*n)!/n!^3)} \\ Seiichi Manyama, Oct 31 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

a(n) = A006480(n-1) + A006480(n) for n > 0. - Seiichi Manyama, Oct 31 2020

Extensions

More terms from Vaclav Kotesovec, Oct 23 2020