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A140802 a(n) = binomial(n+3, 3)*8^n.

%I A140802 #25 Aug 28 2022 04:23:03
%S A140802 1,32,640,10240,143360,1835008,22020096,251658240,2768240640,
%T A140802 29527900160,307090161664,3126736191488,31267361914880,
%U A140802 307863255777280,2990671627550720,28710447624486912,272749252432625664,2567051787601182720,23959150017611038720
%N A140802 a(n) = binomial(n+3, 3)*8^n.
%C A140802 With a different offset, number of n-permutations (n>=3) of 9 objects: r, s, t, u, v, w, z, x, y with repetition allowed, containing exactly (3) three u's.
%C A140802 Example:
%C A140802 (n=4) a(1)=32
%C A140802 uuur, uuru, uruu, ruuu,
%C A140802 uuus, uusu, usuu, suuu,
%C A140802 uuut, uutu, utuu, tuuu,
%C A140802 uuuv, uuvu, uvuu, vuuu,
%C A140802 uuuw, uuwu, uwuu, wuuu,
%C A140802 uuuz, uuzu, uzuu, zuuu,
%C A140802 uuux, uuxu, uxuu, xuuu,
%C A140802 uuuy, uuyu, uyuu, yuuu
%H A140802 Vincenzo Librandi, <a href="/A140802/b140802.txt">Table of n, a(n) for n = 0..400</a>
%H A140802 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (32,-384,2048,-4096).
%F A140802 G.f.: 1/(1-8*x)^4. - _Vincenzo Librandi_, Oct 16 2011
%F A140802 With offset = 3, e.g.f.: exp(8x)*x^3/3!. - _Geoffrey Critzer_, Oct 03 2013
%F A140802 From _Amiram Eldar_, Aug 28 2022: (Start)
%F A140802 Sum_{n>=0} 1/a(n) = 1176*log(8/7) - 156.
%F A140802 Sum_{n>=0} (-1)^n/a(n) = 1944*log(9/8) - 228. (End)
%p A140802 seq(binomial(n+3,3)*8^n,n=0..19);
%t A140802 nn = 21; Drop[Range[0, nn]!CoefficientList[Series[x^3/3! Exp[8x],{x, 0, nn}], x], 3] (* _Geoffrey Critzer_, Oct 03 2013 *)
%o A140802 (Magma) [8^n* Binomial(n+3, 3): n in [0..20]]; // _Vincenzo Librandi_, Oct 16 2011
%K A140802 nonn,easy
%O A140802 0,2
%A A140802 _Zerinvary Lajos_, Jul 15 2008