OFFSET
0,2
COMMENTS
With a different offset, number of n-permutations (n=5) of 4 objects: u, v, z, x with repetition allowed, containing exactly four (4) u's. Example: a(1)=15 because we have uuuuv uuuvu uuvuu uvuuu vuuuu uuuuz uuuzu uuzuu uzuuu zuuuu uuuux uuuxu uuxuu uxuuu xuuuu. - Zerinvary Lajos, Jun 12 2008
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (15,-90,270,-405,243).
FORMULA
a(n) = 3^n*binomial(n+4, 4) = 3^n*A000332(n+4).
a(n) = A027465(n+5, 5).
G.f.: 1/(1-3*x)^5.
E.g.f.: (1/8)*(8 +96*x +216*x^2 +144*x^3 +27*x^4)*exp(3*x). - G. C. Greubel, May 19 2021
From Amiram Eldar, Sep 22 2022: (Start)
Sum_{n>=0} 1/a(n) = 40 - 96*log(3/2).
Sum_{n>=0} (-1)^n/a(n) = 768*log(4/3) - 220. (End)
MAPLE
seq(3^n*binomial(n+4, 4), n=0..30); # Zerinvary Lajos, Jun 12 2008
MATHEMATICA
CoefficientList[Series[1/(1-3x)^5, {x, 0, 30}], x] (* Harvey P. Dale, Jun 13 2017 *)
PROG
(Sage) [3^n*binomial(n+4, 4) for n in range(30)] # Zerinvary Lajos, Mar 10 2009
(Magma) [3^n* Binomial(n+4, 4): n in [0..30]]; // Vincenzo Librandi, Oct 14 2011
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
STATUS
approved