OFFSET
0,2
COMMENTS
With a different offset, number of n-permutations of 8 objects:r,s,t,u,v,z,x,y with repetition allowed, containing exactly five (5) u's. Example: a(1)=42 because we have
uuuuur, uuuuru, uuuruu, uuruuu, uruuuu, ruuuuu
uuuuus, uuuusu, uuusuu, uusuuu, usuuuu, suuuuu,
uuuuut, uuuutu, uuutuu, uutuuu, utuuuu, tuuuuu,
uuuuuv, uuuuvu, uuuvuu, uuvuuu, uvuuuu, vuuuuu,
uuuuuz, uuuuzu, uuuzuu, uuzuuu, uzuuuu, zuuuuu,
uuuuux, uuuuxu, uuuxuu, uuxuuu, uxuuuu, xuuuuu,
uuuuuy, uuuuyu, uuuyuu, uuyuuu, uyuuuu, yuuuuu.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (42,-735,6860,-36015,100842,-117649).
FORMULA
G.f.: 1/(1-7*x)^6. - Zerinvary Lajos, Aug 06 2008
a(n) = 42*a(n-1) - 735*a(n-2) + 6860*a(n-3) - 36015*a(n-4) + 100842*a(n-5) - 117649*a(n-6). - Harvey P. Dale, Sep 08 2011
From Amiram Eldar, Aug 28 2022: (Start)
Sum_{n>=0} 1/a(n) = 45360*log(7/6) - 27965/4.
Sum_{n>=0} (-1)^n/a(n) = 143360*log(8/7) - 229705/12. (End)
MAPLE
seq(binomial(n+5, 5)*7^n, n=0..17);
MATHEMATICA
Table[Binomial[n+5, 5]7^n, {n, 0, 20}] (* or *) LinearRecurrence[ {42, -735, 6860, -36015, 100842, -117649}, {1, 42, 1029, 19208, 302526, 4235364}, 21] (* Harvey P. Dale, Sep 08 2011 *)
PROG
(Magma) [7^n* Binomial(n+5, 5): n in [0..20]]; // Vincenzo Librandi, Oct 12 2011
(PARI) a(n)=binomial(n+5, 5)*7^n \\ Charles R Greathouse IV, Oct 07 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Zerinvary Lajos, Jun 16 2008
STATUS
approved