A140802 a(n) = binomial(n+3, 3)*8^n.

1, 32, 640, 10240, 143360, 1835008, 22020096, 251658240, 2768240640, 29527900160, 307090161664, 3126736191488, 31267361914880, 307863255777280, 2990671627550720, 28710447624486912, 272749252432625664, 2567051787601182720, 23959150017611038720

Offset: 0

Author

Zerinvary Lajos, Jul 15 2008

Keywords

• nonn
• easy

Comments

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.

Example:

(n=4) a(1)=32

uuur, uuru, uruu, ruuu,

uuus, uusu, usuu, suuu,

uuut, uutu, utuu, tuuu,

uuuv, uuvu, uvuu, vuuu,

uuuw, uuwu, uwuu, wuuu,

uuuz, uuzu, uzuu, zuuu,

uuux, uuxu, uxuu, xuuu,

uuuy, uuyu, uyuu, yuuu

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..400
• Index entries for linear recurrences with constant coefficients, signature (32,-384,2048,-4096).

Programs

Magma

[8^n* Binomial(n+3, 3): n in [0..20]]; // Vincenzo Librandi, Oct 16 2011

Maple

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

Mathematica

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

Formula

G.f.: 1/(1-8*x)^4. - Vincenzo Librandi, Oct 16 2011

With offset = 3, e.g.f.: exp(8x)*x^3/3!. - Geoffrey Critzer, Oct 03 2013

From Amiram Eldar, Aug 28 2022: (Start)

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

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