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

1, 36, 3240, 362880, 44906400, 5884534656, 800296713216, 111714888130560, 15898425017080320, 2296439169133824000, 335647548960599715840, 49531592018516268810240, 7367824312754294985523200, 1103342589983347322447462400, 166176904368920474278821888000

Offset: 0

Author

N. J. A. Sloane, Oct 22 2020

Keywords

• nonn

Comments

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).

Links

• Robert Israel, Table of n, a(n) for n = 0..250

Crossrefs

Cf. A338075, A338337.

Programs

Maple

N:= 25: # for a(0)..a(N)
F:= 1/(1-x-2*y-3*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 24 2020

Mathematica

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 *)

Formula

Conjectures from Robert Israel, Oct 25 2020: (Start)

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

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

a(n) = 6^n * (3*n)! / n!^3. - Vaclav Kotesovec, Oct 28 2020

Extensions

More terms from Vaclav Kotesovec, Oct 23 2020