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%I #40 Apr 17 2022 09:26:07
%S 1,30,525,7000,78750,787500,7218750,61875000,502734375,3910156250,
%T 29326171875,213281250000,1510742187500,10458984375000,70971679687500,
%U 473144531250000,3105010986328125,20091247558593750,128360748291015625,810699462890625000
%N 5-idempotent numbers.
%C 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 Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 91, #43.
%H Vincenzo Librandi, <a href="/A050982/b050982.txt">Table of n, a(n) for n = 5..1000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/IdempotentNumber.html">Idempotent Number</a>.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (30,-375,2500,-9375,18750,-15625).
%F a(n) = C(n, 5)*5^(n-5).
%F G.f.: x^5/(1-5*x)^6. - _Zerinvary Lajos_, Aug 06 2008
%F Sum_{n>=5} 1/a(n) = 6400*log(5/4) - 17125/12.
%F Sum_{n>=5} (-1)^(n+1)/a(n) = 32400*log(6/5) - 23625/4. (End)
%p seq(binomial(n, 5)*5^(n-5), n=5..32); # _Zerinvary Lajos_, Jun 16 2008
%t CoefficientList[Series[1 / (1 - 5 x)^6, {x, 0, 33}], x] (* _Vincenzo Librandi_, Aug 12 2017 *)
%o (PARI) a(n)=binomial(n, 5)*5^(n-5) \\ _Charles R Greathouse IV_, Sep 03 2011
%o (Magma) [Binomial(n, 5)*5^(n-5): n in [5..25]]; // _Vincenzo Librandi_, Aug 12 2017
%Y Cf. A001788, A036216, A040075, A050988, A050989, A000389, A054849, A036219, A045543, A036084, A140404, A000389, A054849, A036219, A045543, A036084, A140404.
%K nonn,easy
%O 5,2
%A _Eric W. Weisstein_
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