cp's OEIS Frontend

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

%I A338075 #36 Oct 31 2020 04:49:53
%S A338075 1,7,96,1770,36330,791406,17909892,416226096,9864584730,237338943270,
%T A338075 5778870222840,142077992254380,3521258757984240,87862829835387600,
%U A338075 2205050763983594400,55615552451285359680,1408840444191389714010,35825204161237194511830,914089586182634239686000
%N A338075 Diagonal terms in the expansion of (1+x*y*z)/(1-x-y-z).
%C A338075 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).
%C A338075 If the numerator is changed to 1, we get A006480.
%C A338075 Suggested by Christol's Conjecture (see reference).
%D A338075 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.
%H A338075 Robert Israel, <a href="/A338075/b338075.txt">Table of n, a(n) for n = 0..300</a>
%H A338075 Y. Abdelaziz, C. Koutschan, and J-M. Maillard, <a href="https://arxiv.org/abs/1912.10259">On Christol's conjecture</a>, arXiv:1912.10259 [math.NT], 2019-2020.
%F A338075 Conjectures from _Robert Israel_, Oct 25 2020: (Start)
%F A338075 G.f.: (x + 1)*LegendreP(-1/3, 1 - 54*x).
%F A338075 (-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)
%F A338075 a(n) = (28*n^2 - 27*n + 6) * (3*n)! / (3 * (3*n - 1) * (3*n - 2) * n!^3). - _Vaclav Kotesovec_, Oct 28 2020
%F A338075 a(n) = A006480(n-1) + A006480(n) for n > 0. - _Seiichi Manyama_, Oct 31 2020
%p A338075 N:= 25: # for a(0)..a(N)
%p A338075 F:=  (1+x*y*z)/(1-x-y-z):
%p A338075 S1:= series(F, x, N+1):
%p A338075 L1:= [seq(coeff(S1, x, i), i=0..N)]:
%p A338075 L2:= [seq(coeff(series(L1[i+1], y, i+1), y, i), i=0..N)]:
%p A338075 seq(coeff(series(L2[i+1], z, i+1), z, i), i=0..N); # _Robert Israel_, Oct 25 2020
%t A338075 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 *)
%o A338075 (PARI) {a(n) = if(n==0, 1, (3*(n-1))!/(n-1)!^3+(3*n)!/n!^3)} \\ _Seiichi Manyama_, Oct 31 2020
%Y A338075 Other examples arising from diagonal terms of multivariate g.f.s: A000172, A006480, A338076.
%K A338075 nonn
%O A338075 0,2
%A A338075 _N. J. A. Sloane_, Oct 22 2020
%E A338075 More terms from _Vaclav Kotesovec_, Oct 23 2020