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A141054 8-idempotent numbers: a(n) = binomial(n+8,8)*8^n.

%I A141054 #22 Apr 18 2022 12:32:08
%S A141054 1,72,2880,84480,2027520,42172416,787218432,13495173120,215922769920,
%T A141054 3262832967680,46984794734592,649244436332544,8656592484433920,
%U A141054 111869810568069120,1406363332855726080,17251390216363573248,207016682596362878976,2435490383486622105600
%N A141054 8-idempotent numbers: a(n) = binomial(n+8,8)*8^n.
%C A141054 With a different offset, number of n-permutations of 9 objects:
%C A141054 p, r, s, t, u, v, z, x, y with repetition allowed, containing exactly eight (8) u's. Example: a(1)=72 because we have
%C A141054 uuuuuuuup, uuuuuuupu, uuuuuupuu, uuuuupuuu, uuuupuuuu, uuupuuuuu, uupuuuuuu, upuuuuuuu, puuuuuuuu,
%C A141054 uuuuuuuur, uuuuuuuru, uuuuuuruu, uuuuuruuu, uuuuruuuu, uuuruuuuu, uuruuuuuu, uruuuuuuu, ruuuuuuuu,
%C A141054 uuuuuuuus, uuuuuuusu, uuuuuusuu, uuuuusuuu, uuuusuuuu, uuusuuuuu, uusuuuuuu, usuuuuuuu, suuuuuuuu,
%C A141054 uuuuuuuut, uuuuuuutu, uuuuuutuu, uuuuutuuu, uuuutuuuu, uuutuuuuu, uutuuuuuu, utuuuuuuu, tuuuuuuuu,
%C A141054 uuuuuuuuv, uuuuuuuvu, uuuuuuvuu, uuuuuvuuu, uuuuvuuuu, uuuvuuuuu, uuvuuuuuu, uvuuuuuuu, vuuuuuuuu,
%C A141054 uuuuuuuuz, uuuuuuuzu, uuuuuuzuu, uuuuuzuuu, uuuuzuuuu, uuuzuuuuu, uuzuuuuuu, uzuuuuuuu, zuuuuuuuu,
%C A141054 uuuuuuuux, uuuuuuuxu, uuuuuuxuu, uuuuuxuuu, uuuuxuuuu, uuuxuuuuu, uuxuuuuuu, uxuuuuuuu, xuuuuuuuu,
%C A141054 uuuuuuuuy, uuuuuuuyu, uuuuuuyuu, uuuuuyuuu, uuuuyuuuu, uuuyuuuuu, uuyuuuuuu, uyuuuuuuu, yuuuuuuuu.
%H A141054 Vincenzo Librandi, <a href="/A141054/b141054.txt">Table of n, a(n) for n = 0..400</a>
%F A141054 a(n) = binomial(n+8,8)*8^n.
%F A141054 G.f.: 1/(1-8*x)^9. - _Vincenzo Librandi_, Oct 16 2011
%F A141054 From _Amiram Eldar_, Apr 17 2022: (Start)
%F A141054 Sum_{n>=0} 1/a(n) = 738990736/105 - 52706752*log(8/7).
%F A141054 Sum_{n>=0} (-1)^n/a(n) = 306110016*log(9/8) - 1261909808/35. (End)
%p A141054 seq(binomial(n+8,8)*8^n, n=0..17);
%t A141054 Table[Binomial[n + 8, 8] 8^n, {n, 0, 15}] (* _Michael De Vlieger_, Jul 24 2017 *)
%o A141054 (Magma) [8^n* Binomial(n+8, 8): n in [0..20]]; // _Vincenzo Librandi_, Oct 16 2011
%o A141054 (PARI) vector(15,n,binomial(n+7,8)*8^(n-1)) \\ _Derek Orr_, Jul 24 2017
%Y A141054 Cf. A000027, A001788, A036216, A040075, A050982, A050988, A050989, A059297, A059300.
%K A141054 nonn,easy
%O A141054 0,2
%A A141054 _Zerinvary Lajos_, Aug 01 2008