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A050982 5-idempotent numbers.

%I A050982 #43 Feb 16 2025 08:32:41
%S A050982 1,30,525,7000,78750,787500,7218750,61875000,502734375,3910156250,
%T A050982 29326171875,213281250000,1510742187500,10458984375000,70971679687500,
%U A050982 473144531250000,3105010986328125,20091247558593750,128360748291015625,810699462890625000
%N A050982 5-idempotent numbers.
%C A050982 Number of n-permutations of 6 objects: t,u,v,z,x, y with repetition allowed, containing exactly five u's. Example: a(6)=30 because we have uuuuut, uuuutu, uuutuu, uutuuu, utuuuu, tuuuuu, uuuuuv, uuuuvu, uuuvuu, uuvuuu, uvuuuu, vuuuuu, uuuuuz, uuuuzu, uuuzuu, uuzuuu, uzuuuu, zuuuuu, uuuuux, uuuuxu, uuuxuu, uuxuuu, uxuuuu, xuuuuu, uuuuuy, uuuuyu, uuuyuu, uuyuuu, uyuuuu, yuuuuu. - _Zerinvary Lajos_, Jun 16 2008
%D A050982 Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 91, #43.
%H A050982 Vincenzo Librandi, <a href="/A050982/b050982.txt">Table of n, a(n) for n = 5..1000</a>
%H A050982 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/IdempotentNumber.html">Idempotent Number</a>.
%H A050982 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (30,-375,2500,-9375,18750,-15625).
%F A050982 a(n) = C(n, 5)*5^(n-5).
%F A050982 G.f.: x^5/(1-5*x)^6. - _Zerinvary Lajos_, Aug 06 2008
%F A050982 From _Amiram Eldar_, Apr 17 2022: (Start)
%F A050982 Sum_{n>=5} 1/a(n) = 6400*log(5/4) - 17125/12.
%F A050982 Sum_{n>=5} (-1)^(n+1)/a(n) = 32400*log(6/5) - 23625/4. (End)
%p A050982 seq(binomial(n, 5)*5^(n-5), n=5..32); # _Zerinvary Lajos_, Jun 16 2008
%t A050982 CoefficientList[Series[1 / (1 - 5 x)^6, {x, 0, 33}], x] (* _Vincenzo Librandi_, Aug 12 2017 *)
%o A050982 (PARI) a(n)=binomial(n, 5)*5^(n-5) \\ _Charles R Greathouse IV_, Sep 03 2011
%o A050982 (Magma) [Binomial(n, 5)*5^(n-5): n in [5..25]]; // _Vincenzo Librandi_, Aug 12 2017
%Y A050982 Cf. A001788, A036216, A040075, A050988, A050989, A000389, A054849, A036219, A045543, A036084, A140404, A000389, A054849, A036219, A045543, A036084, A140404.
%K A050982 nonn,easy
%O A050982 5,2
%A A050982 _Eric W. Weisstein_