a(n) = (5*8^(n+4) - 12*7^(n+4) + 20*5^(n+4) - 15*4^(n+4) +2*2^(n+4))/360.
In general, for the expansion of 1/((1-r*x)(1-s*x)(1-t*x)(1-u*x)(1-v*x)) with v > u > t > s > r , we have the formula
a(n) = (K*v^(n+4) - L*u^(n+4) + M*t^(n+4) - N*s^(n+4) + P*r^(n+4)) / (K*L*M*N*P)^(1/3) where K,L,M,N,P have the following values:
K = (u-t)*(u-s)*(u-r)*(t-s)*(t-r)*(s-r);
L = (v-t)*(v-s)*(v-r)*(t-s)*(t-r)*(s-r);
M = (v-u)*(v-s)*(v-r)*(u-s)*(u-r)*(s-r);
N = (v-u)*(v-t)*(v-r)*(u-t)*(u-r)*(t-r);
P = (v-u)*(v-t)*(v-s)*(u-t)*(u-s)*(t-s).
Directly using the formula we obtain a(n) = (180*8^(n+4) - 432*7^(n+4) + 720*5^(n+4) - 540*4^(n+4) + 72*2^(n+4))/12960 simplifies after by 36.
