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

1, 72, 2880, 84480, 2027520, 42172416, 787218432, 13495173120, 215922769920, 3262832967680, 46984794734592, 649244436332544, 8656592484433920, 111869810568069120, 1406363332855726080, 17251390216363573248, 207016682596362878976, 2435490383486622105600

Offset

0,2

Comments

With a different offset, number of n-permutations of 9 objects:

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

uuuuuuuup, uuuuuuupu, uuuuuupuu, uuuuupuuu, uuuupuuuu, uuupuuuuu, uupuuuuuu, upuuuuuuu, puuuuuuuu,

uuuuuuuur, uuuuuuuru, uuuuuuruu, uuuuuruuu, uuuuruuuu, uuuruuuuu, uuruuuuuu, uruuuuuuu, ruuuuuuuu,

uuuuuuuus, uuuuuuusu, uuuuuusuu, uuuuusuuu, uuuusuuuu, uuusuuuuu, uusuuuuuu, usuuuuuuu, suuuuuuuu,

uuuuuuuut, uuuuuuutu, uuuuuutuu, uuuuutuuu, uuuutuuuu, uuutuuuuu, uutuuuuuu, utuuuuuuu, tuuuuuuuu,

uuuuuuuuv, uuuuuuuvu, uuuuuuvuu, uuuuuvuuu, uuuuvuuuu, uuuvuuuuu, uuvuuuuuu, uvuuuuuuu, vuuuuuuuu,

uuuuuuuuz, uuuuuuuzu, uuuuuuzuu, uuuuuzuuu, uuuuzuuuu, uuuzuuuuu, uuzuuuuuu, uzuuuuuuu, zuuuuuuuu,

uuuuuuuux, uuuuuuuxu, uuuuuuxuu, uuuuuxuuu, uuuuxuuuu, uuuxuuuuu, uuxuuuuuu, uxuuuuuuu, xuuuuuuuu,

uuuuuuuuy, uuuuuuuyu, uuuuuuyuu, uuuuuyuuu, uuuuyuuuu, uuuyuuuuu, uuyuuuuuu, uyuuuuuuu, yuuuuuuuu.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..400

Formula

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

G.f.: 1/(1-8*x)^9. - Vincenzo Librandi, Oct 16 2011

From Amiram Eldar, Apr 17 2022: (Start)

Sum_{n>=0} 1/a(n) = 738990736/105 - 52706752*log(8/7).

Sum_{n>=0} (-1)^n/a(n) = 306110016*log(9/8) - 1261909808/35. (End)

Maple

seq(binomial(n+8, 8)*8^n, n=0..17);

Mathematica

Table[Binomial[n + 8, 8] 8^n, {n, 0, 15}] (* Michael De Vlieger, Jul 24 2017 *)

Prog

(Magma) [8^n* Binomial(n+8, 8): n in [0..20]]; // Vincenzo Librandi, Oct 16 2011
(PARI) vector(15, n, binomial(n+7, 8)*8^(n-1)) \\ Derek Orr, Jul 24 2017

Crossrefs

Cf. A000027, A001788, A036216, A040075, A050982, A050988, A050989, A059297, A059300.

Sequence in context: A035731 A035803 A017735 * A173192 A004366 A004389

Adjacent sequences: A141051 A141052 A141053 * A141055 A141056 A141057

Keyword

nonn,easy

Author

Zerinvary Lajos, Aug 01 2008

Status

approved