A140802 - OEIS

%I #25 Aug 28 2022 04:23:03

%S 1,32,640,10240,143360,1835008,22020096,251658240,2768240640,

%T 29527900160,307090161664,3126736191488,31267361914880,

%U 307863255777280,2990671627550720,28710447624486912,272749252432625664,2567051787601182720,23959150017611038720

%N a(n) = binomial(n+3, 3)*8^n.

%C With a different offset, number of n-permutations (n>=3) of 9 objects: r, s, t, u, v, w, z, x, y with repetition allowed, containing exactly (3) three u's.

%C Example:

%C (n=4) a(1)=32

%C uuur, uuru, uruu, ruuu,

%C uuus, uusu, usuu, suuu,

%C uuut, uutu, utuu, tuuu,

%C uuuv, uuvu, uvuu, vuuu,

%C uuuw, uuwu, uwuu, wuuu,

%C uuuz, uuzu, uzuu, zuuu,

%C uuux, uuxu, uxuu, xuuu,

%C uuuy, uuyu, uyuu, yuuu

%H Vincenzo Librandi, <a href="/A140802/b140802.txt">Table of n, a(n) for n = 0..400</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (32,-384,2048,-4096).

%F G.f.: 1/(1-8*x)^4. - _Vincenzo Librandi_, Oct 16 2011

%F With offset = 3, e.g.f.: exp(8x)*x^3/3!. - _Geoffrey Critzer_, Oct 03 2013

%F From _Amiram Eldar_, Aug 28 2022: (Start)

%F Sum_{n>=0} 1/a(n) = 1176*log(8/7) - 156.

%F Sum_{n>=0} (-1)^n/a(n) = 1944*log(9/8) - 228. (End)

%p seq(binomial(n+3,3)*8^n,n=0..19);

%t nn = 21; Drop[Range[0, nn]!CoefficientList[Series[x^3/3! Exp[8x],{x, 0, nn}], x], 3] (* _Geoffrey Critzer_, Oct 03 2013 *)

%o (Magma) [8^n* Binomial(n+3, 3): n in [0..20]]; // _Vincenzo Librandi_, Oct 16 2011

%K nonn,easy

%O 0,2

%A _Zerinvary Lajos_, Jul 15 2008