OFFSET
0,2
COMMENTS
With a different offset, number of n-permutations (n>=7) of 4 objects: u, v, z, x with repetition allowed, containing exactly seven (7) u's. Example: a(1)=24 because we have uuuuuuuv, uuuuuuuz, uuuuuuux, uuuuuuvu, uuuuuuzu, uuuuuuxu, uuuuuvuu, uuuuuzuu, uuuuuxuu, uuuuvuuu, uuuuzuuu, uuuuxuuu, uuuvuuuu, uuuzuuuu, uuuxuuuu, uuvuuuuu, uuzuuuuu, uuxuuuuu, uvuuuuuu, uzuuuuuu, uxuuuuuu, vuuuuuuu, zuuuuuuu, xuuuuuuu. - Zerinvary Lajos, Jun 23 2008
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (24,-252,1512,-5670,13608,-20412,17496,-6561).
FORMULA
a(n) = 3^n*binomial(n+7, 7).
a(n) = A027465(n+8, 8.)
G.f.: 1/(1-3*x)^8.
E.g.f.: (1/560)*(560 +11760*x +52920*x^2 +88200*x^3 +66150*x^4 +23814*x^5 +3969*x^6 +243*x^7)*exp(3*x). - G. C. Greubel, May 19 2021
From Amiram Eldar, Sep 22 2022: (Start)
Sum_{n>=0} 1/a(n) = 1344*log(3/2) - 5439/10.
Sum_{n>=0} (-1)^n/a(n) = 86016*log(4/3) - 247443/10. (End)
MAPLE
seq(3^n*binomial(n+7, 7), n=0..30); # Zerinvary Lajos, Jun 23 2008
MATHEMATICA
Table[3^n*Binomial[n+7, 7], {n, 0, 30}] (* G. C. Greubel, May 19 2021 *)
PROG
(Sage) [3^n*binomial(n+7, 7) for n in range(30)] # Zerinvary Lajos, Mar 13 2009
(Magma) [3^n*Binomial(n+7, 7): n in [0..30]]; // Vincenzo Librandi, Oct 15 2011
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
STATUS
approved