OFFSET
0,2
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).
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)
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 *)
PROG
(Magma) [8^n* Binomial(n+6, 6): n in [0..20]]; // Vincenzo Librandi, Oct 16 2011
(PARI) a(n)=binomial(n+6, 6)<<(3*n) \\ Charles R Greathouse IV, Dec 15 2011
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Zerinvary Lajos, Jun 16 2008
STATUS
approved