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A140405 a(n) = binomial(n+6, 6)*5^n.

%I A140405 #19 Aug 29 2022 04:40:11
%S A140405 1,35,700,10500,131250,1443750,14437500,134062500,1173046875,
%T A140405 9775390625,78203125000,604296875000,4532226562500,33120117187500,
%U A140405 236572265625000,1656005859375000,11385040283203125,77016448974609375,513442993164062500,3377914428710937500
%N A140405 a(n) = binomial(n+6, 6)*5^n.
%C A140405 With a different offset, number of n-permutations (n>=6) of 6 objects: t, u, v, z, x, y with repetition allowed, containing exactly six (6) u's.
%C A140405 If n=6 then a(0)=1.
%C A140405 Example: a(1)=35 because we have
%C A140405 uuuuuut, uuuuutu, uuuutuu, uuutuuu, uutuuuu, utuuuuu, tuuuuuu,
%C A140405 uuuuuuv, uuuuuvu, uuuuvuu, uuuvuuu, uuvuuuu, uvuuuuu, vuuuuuu,
%C A140405 uuuuuuz, uuuuuzu, uuuuzuu, uuuzuuu, uuzuuuu, uzuuuuu, zuuuuuu,
%C A140405 uuuuuux, uuuuuxu, uuuuxuu, uuuxuuu, uuxuuuu, uxuuuuu, xuuuuuu,
%C A140405 uuuuuuy, uuuuuyu, uuuuyuu, uuuyuuu, uuyuuuu, uyuuuuu, yuuuuuu.
%H A140405 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (35,-525,4375,-21875,65625,-109375,78125).
%F A140405 G.f.: 1/(1-5*x)^7. - _Zerinvary Lajos_, Aug 06 2008
%F A140405 From _Amiram Eldar_, Aug 29 2022: (Start)
%F A140405 Sum_{n>=0} 1/a(n) = 6856 - 30720*log(5/4).
%F A140405 Sum_{n>=0} (-1)^n/a(n) = 233280*log(6/5) - 42531. (End)
%p A140405 seq(binomial(n+6,6)*5^n,n=0..18);
%t A140405 Table[Binomial[n+6,6]5^n,{n,0,20}] (* _Harvey P. Dale_, Dec 03 2017 *)
%Y A140405 Cf. A000579, A002409, A036220, A036226, A050988, A054337, A140406.
%K A140405 nonn,easy
%O A140405 0,2
%A A140405 _Zerinvary Lajos_, Jun 16 2008