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

Author

Zerinvary Lajos, Aug 01 2008

Keywords

• nonn
• easy

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

Crossrefs

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

Programs

Magma

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

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 *)

PARI

vector(15,n,binomial(n+7,8)*8^(n-1)) \\ Derek Orr, Jul 24 2017

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)