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A140520 a(n) = binomial(n+9, 9)*5^n.

%I A140520 #23 Aug 29 2022 04:40:26
%S A140520 1,50,1375,27500,446875,6256250,78203125,893750000,9496093750,
%T A140520 94960937500,902128906250,8201171875000,71760253906250,
%U A140520 607202148437500,4987731933593750,39901855468750000,311733245849609375,2383842468261718750,17878818511962890625,131738662719726562500
%N A140520 a(n) = binomial(n+9, 9)*5^n.
%C A140520 With a different offset, number of n-permutations (n>=9) of 6 objects: t, u, v, z, x, y with repetition allowed, containing exactly nine (9) u's.
%C A140520 Example: a(1)=50 because we have
%C A140520 uuuuuuuuut, uuuuuuuuuv, uuuuuuuuuz, uuuuuuuuux, uuuuuuuuuy,
%C A140520 uuuuuuuutu, uuuuuuuuvu, uuuuuuuuzu, uuuuuuuuxu, uuuuuuuuyu,
%C A140520 uuuuuuutuu, uuuuuuuvuu, uuuuuuuzuu, uuuuuuuxuu, uuuuuuuyuu,
%C A140520 uuuuuutuuu, uuuuuuvuuu, uuuuuuzuuu, uuuuuuxuuu, uuuuuuyuuu,
%C A140520 uuuuutuuuu, uuuuuvuuuu, uuuuuzuuuu, uuuuuxuuuu, uuuuuyuuuu,
%C A140520 uuuutuuuuu, uuuuvuuuuu, uuuuzuuuuu, uuuuxuuuuu, uuuuyuuuuu,
%C A140520 uuutuuuuuu, uuuvuuuuuu, uuuzuuuuuu, uuuxuuuuuu, uuuyuuuuuu,
%C A140520 uutuuuuuuu, uuvuuuuuuu, uuzuuuuuuu, uuxuuuuuuu, uuyuuuuuuu,
%C A140520 utuuuuuuuu, uvuuuuuuuu, uzuuuuuuuu, uxuuuuuuuu. uyuuuuuuuu,
%C A140520 tuuuuuuuuu, vuuuuuuuuu, zuuuuuuuuu, xuuuuuuuuu. yuuuuuuuuu.
%H A140520 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (50,-1125,15000,-131250,787500,-3281250,9375000,-17578125,19531250,-9765625).
%F A140520 From _Chai Wah Wu_, Mar 20 2017: (Start)
%F A140520 a(n) = 50*a(n-1) - 1125*a(n-2) + 15000*a(n-3) - 131250*a(n-4) + 787500*a(n-5) - 3281250*a(n-6) + 9375000*a(n-7) - 17578125*a(n-8) + 19531250*a(n-9) - 9765625*a(n-10) for n > 9.
%F A140520 G.f.: 1/(1 - 5*x)^10. (End)
%F A140520 From _Amiram Eldar_, Aug 29 2022: (Start)
%F A140520 Sum_{n>=0} 1/a(n) = 2949120*log(5/4) - 36852261/56.
%F A140520 Sum_{n>=0} (-1)^n/a(n) = 75582720*log(6/5) - 771700059/56. (End)
%p A140520 seq(binomial(n+9,9)*5^n,n=0..20);
%t A140520 Table[Binomial[n + 9, 9] 5^n, {n, 0, 16}] (* or *)
%t A140520 CoefficientList[Series[1/(1 - 5 x)^10, {x, 0, 16}], x] (* _Michael De Vlieger_, Mar 20 2017 *)
%K A140520 nonn,easy
%O A140520 0,2
%A A140520 _Zerinvary Lajos_, Jul 02 2008