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A140346 a(n) = binomial(n+8, 8)*5^n.

%I A140346 #22 Aug 29 2022 04:40:00
%S A140346 1,45,1125,20625,309375,4021875,46921875,502734375,5027343750,
%T A140346 47480468750,427324218750,3690527343750,30754394531250,
%U A140346 248400878906250,1951721191406250,14963195800781250,112223968505859375,825176239013671875,5959606170654296875,42344570159912109375
%N A140346 a(n) = binomial(n+8, 8)*5^n.
%C A140346 With a different offset, number of n-permutations (n>=8) of 6 objects: t, u, v, z, x, y with repetition allowed, containing exactly eight (8) u's.
%C A140346 If n=8 then a(0)=1.
%C A140346 Example: a(1)=45 because we have
%C A140346 uuuuuuuut, uuuuuuuuv, uuuuuuuuz, uuuuuuuux, uuuuuuuuy,
%C A140346 uuuuuuutu, uuuuuuuvu, uuuuuuuzu, uuuuuuuxu, uuuuuuuyu,
%C A140346 uuuuuutuu, uuuuuuvuu, uuuuuuzuu, uuuuuuxuu, uuuuuuyuu,
%C A140346 uuuuutuuu, uuuuuvuuu, uuuuuzuuu, uuuuuxuuu, uuuuuyuuu,
%C A140346 uuuutuuuu, uuuuvuuuu, uuuuzuuuu, uuuuxuuuu, uuuuyuuuu,
%C A140346 uuutuuuuu, uuuvuuuuu, uuuzuuuuu, uuuxuuuuu, uuuyuuuuu,
%C A140346 uutuuuuuu, uuvuuuuuu, uuzuuuuuu, uuxuuuuuu, uuyuuuuuu,
%C A140346 utuuuuuuu, uvuuuuuuu, uzuuuuuuu, uxuuuuuuu, uyuuuuuuu,
%C A140346 tuuuuuuuu, vuuuuuuuu, zuuuuuuuu, xuuuuuuuu, yuuuuuuuu.
%H A140346 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (45,-900,10500,-78750,393750,-1312500,2812500,-3515625,1953125).
%F A140346 From _Chai Wah Wu_, Mar 20 2017: (Start)
%F A140346 a(n) = 45*a(n-1) - 900*a(n-2) + 10500*a(n-3) - 78750*a(n-4) + 393750*a(n-5) - 1312500*a(n-6) + 2812500*a(n-7) - 3515625*a(n-8) + 1953125*a(n-9) for n > 8.
%F A140346 G.f.: 1/(1 - 5*x)^9. (End)
%F A140346 From _Amiram Eldar_, Aug 29 2022: (Start)
%F A140346 Sum_{n>=0} 1/a(n) = 11197440*log(6/5) - 14290736/7.
%F A140346 Sum_{n>=0} (-1)^n/a(n) = 3071048/21 - 655360*log(5/4). (End)
%p A140346 seq(binomial(n+8,8)*5^n,n=0..18);
%t A140346 Table[Binomial[n + 8, 8] 5^n, {n, 0, 16}] (* or *)
%t A140346 CoefficientList[Series[1/(1 - 5 x)^9, {x, 0, 16}], x] (* _Michael De Vlieger_, Mar 20 2017 *)
%K A140346 nonn,easy
%O A140346 0,2
%A A140346 _Zerinvary Lajos_, Jun 23 2008