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A229025 Expansion of 1/((1-2x)(1-4x)(1-5x)(1-7x)(1-8x)).

%I A229025 #18 Jun 11 2023 10:29:09
%S A229025 1,26,417,5334,59829,616602,5996089,55931678,505925277,4470884418,
%T A229025 38804443041,332070466182,2809908472645,23562316644074,
%U A229025 196128590350473,1622684846427246,13358308842842733,109510501632625170,894623468678425585,7286920685445869270
%N A229025 Expansion of 1/((1-2x)(1-4x)(1-5x)(1-7x)(1-8x)).
%C A229025 This sequence was chosen to illustrate a method of solution.
%H A229025 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (26, -259, 1226, -2728, 2240).
%F A229025 a(n) = (5*8^(n+4) - 12*7^(n+4) + 20*5^(n+4) - 15*4^(n+4) +2*2^(n+4))/360.
%F A229025 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
%F A229025 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:
%F A229025 K = (u-t)*(u-s)*(u-r)*(t-s)*(t-r)*(s-r);
%F A229025 L = (v-t)*(v-s)*(v-r)*(t-s)*(t-r)*(s-r);
%F A229025 M = (v-u)*(v-s)*(v-r)*(u-s)*(u-r)*(s-r);
%F A229025 N = (v-u)*(v-t)*(v-r)*(u-t)*(u-r)*(t-r);
%F A229025 P = (v-u)*(v-t)*(v-s)*(u-t)*(u-s)*(t-s).
%F A229025 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.
%t A229025 nn = 20; CoefficientList[Series[1/((1 - 2*x) (1 - 4*x) (1 - 5*x) (1 - 7*x) (1 - 8*x)), {x, 0, nn}], x] (* _T. D. Noe_, Sep 12 2013 *)
%K A229025 nonn,easy
%O A229025 0,2
%A A229025 _Yahia Kahloune_, Sep 11 2013