A140406 a(n) = binomial(n+6, 6)*8^n.

1, 56, 1792, 43008, 860160, 15138816, 242221056, 3598712832, 50381979648, 671759728640, 8598524526592, 106309030510592, 1275708366127104, 14915974742409216, 170468282770391040, 1909244767028379648, 21001692437312176128, 227312435792084729856

Offset: 0

Author

Zerinvary Lajos, Jun 16 2008

Keywords

• nonn
• easy

Comments

With a different offset, number of n-permutations (n >= 6) of 9 objects: p, r, s, t, u, v, z, x, y with repetition allowed, containing exactly six (6) u's.

If n=6 then a(0)=1.

Example: a(1)=56 because we have

uuuuuup, uuuuupu, uuuupuu, uuupuuu, uupuuuu, upuuuuu, puuuuuu,

uuuuuur, uuuuuru, uuuuruu, uuuruuu, uuruuuu, uruuuuu, ruuuuuu,

uuuuuus, uuuuusu, uuuusuu, uuusuuu, uusuuuu, usuuuuu, suuuuuu,

uuuuuut, uuuuutu, uuuutuu, uuutuuu, uutuuuu, utuuuuu, tuuuuuu,

uuuuuuv, uuuuuvu, uuuuvuu, uuuvuuu, uuvuuuu, uvuuuuu, vuuuuuu,

uuuuuuz, uuuuuzu, uuuuzuu, uuuzuuu, uuzuuuu, uzuuuuu, zuuuuuu,

uuuuuux, uuuuuxu, uuuuxuu, uuuxuuu, uuxuuuu, uxuuuuu, xuuuuuu,

uuuuuuy, uuuuuyu, uuuuyuu, uuuyuuu, uuyuuuu, uyuuuuu, yuuuuuu.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..400
• Index entries for linear recurrences with constant coefficients, signature (56,-1344,17920,-143360,688128,-1835008,2097152).

Crossrefs

Cf. A000579, A002409, A036220, A036226, A050988, A054337, A140405.

Programs

Magma

[8^n* Binomial(n+6, 6): n in [0..20]]; // Vincenzo Librandi, Oct 16 2011

Maple

seq(binomial(n+6,6)*8^n,n=0..17);

Mathematica

Table[Binomial[n+6,6]8^n,{n,0,20}] (* or *) LinearRecurrence[ {56,-1344,17920,-143360,688128,-1835008,2097152},{1,56,1792,43008,860160,15138816,242221056},20] (* Harvey P. Dale, Dec 15 2011 *)

PARI

a(n)=binomial(n+6,6)<<(3*n) \\ Charles R Greathouse IV, Dec 15 2011

Formula

G.f.: 1/(1-8*x)^7. - Zerinvary Lajos, Aug 06 2008

a(n) = 56*a(n-1) - 1344*a(n-2) + 17920*a(n-3) - 143360*a(n-4) + 688128*a(n-5) - 1835008*a(n-6) + 2097152*a(n-7). - Harvey P. Dale, Dec 15 2011

From Amiram Eldar, Aug 28 2022: (Start)

Sum_{n>=0} 1/a(n) = 538628/5 - 806736*log(8/7).

Sum_{n>=0} (-1)^n/a(n) = 2834352*log(9/8) - 1669188/5. (End)