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A140220 a(n) = binomial(n+7, 7)*5^n.

%I A140220 #24 Aug 29 2022 04:39:37
%S A140220 1,40,900,15000,206250,2475000,26812500,268125000,2513671875,
%T A140220 22343750000,189921875000,1553906250000,12301757812500,94628906250000,
%U A140220 709716796875000,5204589843750000,37407989501953125,264056396484375000,1833724975585937500,12546539306640625000
%N A140220 a(n) = binomial(n+7, 7)*5^n.
%C A140220 With a different offset, number of n-permutations (n>=7) of 6 objects: t, u, v, z, x, y with repetition allowed, containing exactly seven (7) u's.
%C A140220 If n=7 then a(0)=1.
%C A140220 Example: a(1)=40 because we have
%C A140220 uuuuuuut, uuuuuuuv, uuuuuuuz, uuuuuuux, uuuuuuuy,
%C A140220 uuuuuutu, uuuuuuvu, uuuuuuzu, uuuuuuxu, uuuuuuyu,
%C A140220 uuuuutuu, uuuuuvuu, uuuuuzuu, uuuuuxuu, uuuuuyuu,
%C A140220 uuuutuuu, uuuuvuuu, uuuuzuuu, uuuuxuuu, uuuuyuuu,
%C A140220 uuutuuuu, uuuvuuuu, uuuzuuuu, uuuxuuuu, uuuyuuuu,
%C A140220 uutuuuuu, uuvuuuuu, uuzuuuuu, uuxuuuuu, uuyuuuuu,
%C A140220 utuuuuuu, uvuuuuuu, uzuuuuuu, uxuuuuuu, uyuuuuuu,
%C A140220 tuuuuuuu, vuuuuuuu, zuuuuuuu, xuuuuuuu, yuuuuuuu.
%H A140220 Vincenzo Librandi, <a href="/A140220/b140220.txt">Table of n, a(n) for n = 0..1000</a>
%H A140220 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (40,-700,7000,-43750,175000,-437500,625000,-390625).
%F A140220 From _Chai Wah Wu_, Mar 20 2017: (Start)
%F A140220 a(n) = 40*a(n-1) - 700*a(n-2) + 7000*a(n-3) - 43750*a(n-4) + 175000*a(n-5) - 437500*a(n-6) + 625000*a(n-7) - 390625*a(n-8) for n > 7.
%F A140220 G.f.: 1/(1 - 5*x)^8. (End)
%F A140220 From _Amiram Eldar_, Aug 29 2022: (Start)
%F A140220 Sum_{n>=0} 1/a(n) = 143360*log(5/4) - 191933/6.
%F A140220 Sum_{n>=0} (-1)^n/a(n) = 1632960*log(6/5) - 1786337/6. (End)
%p A140220 seq(binomial(n+7,7)*5^n,n=0..18);
%t A140220 Table[Binomial[n+7,7]*5^n,{n,0,20}] (* _Harvey P. Dale_, Oct 07 2014 *)
%t A140220 CoefficientList[Series[1 / (1 - 5 x)^8, {x, 0, 33}], x] (* _Vincenzo Librandi_, Feb 09 2018 *)
%o A140220 (Magma) [Binomial(n+7, 7)*5^n: n in [0..20]]; // _Vincenzo Librandi_, Feb 09 2018
%K A140220 nonn,easy
%O A140220 0,2
%A A140220 _Zerinvary Lajos_, Jun 23 2008