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A338076 Diagonal terms in the expansion of 1/(1-x-2*y-3*z). 3

%I

%S 1,36,3240,362880,44906400,5884534656,800296713216,111714888130560,

%T 15898425017080320,2296439169133824000,335647548960599715840,

%U 49531592018516268810240,7367824312754294985523200,1103342589983347322447462400,166176904368920474278821888000

%N Diagonal terms in the expansion of 1/(1-x-2*y-3*z).

%C Expand the rational function 1/(1-x-2*y-3*z) as Sum_i Sum_j Sum_k c(i,j,k)*x^i*y^j*z^k; a(n) = c(n,n,n).

%H Robert Israel, <a href="/A338076/b338076.txt">Table of n, a(n) for n = 0..250</a>

%F Conjectures from _Robert Israel_, Oct 25 2020: (Start)

%F a(n+1) = 18*(3*n+1)*(3*n+2)*a(n)/(n+1)^2.

%F G.f.: hypergeom([1/3, 2/3], [1], 162*x). (End)

%F a(n) = 6^n * (3*n)! / n!^3. - _Vaclav Kotesovec_, Oct 28 2020

%p N:= 25: # for a(0)..a(N)

%p F:= 1/(1-x-2*y-3*z):

%p S1:= series(F,x,N+1):

%p L1:= [seq(coeff(S1,x,i),i=0..N)]:

%p L2:= [seq(coeff(series(L1[i+1],y,i+1),y,i),i=0..N)]:

%p seq(coeff(series(L2[i+1],z,i+1),z,i),i=0..N); # _Robert Israel_, Oct 24 2020

%t nmax = 20; Flatten[{1, Table[Coefficient[Series[1/(1-x-2*y-3*z), {x, 0, n}, {y, 0, n}, {z, 0, n}], x^n*y^n*z^n], {n, 1, nmax}]}] (* _Vaclav Kotesovec_, Oct 23 2020 *)

%Y Cf. A338075, A338337.

%K nonn

%O 0,2

%A _N. J. A. Sloane_, Oct 22 2020

%E More terms from _Vaclav Kotesovec_, Oct 23 2020

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Last modified February 7 02:40 EST 2023. Contains 360111 sequences. (Running on oeis4.)