OFFSET
0,2
COMMENTS
With a different offset, number of n-permutations (n>=6) of 5 objects: u, v, z, x, y with repetition allowed, containing exactly six (6) u's. Example: a(1)=28 because we have 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. - Zerinvary Lajos, Jun 16 2008
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..400
FORMULA
a(n) = binomial(n+6, 6)*4^n.
G.f.: 1/(1 - 4*x)^7.
a(n) = A054335(n+13, 13).
E.g.f.: (45 + 1080*x + 5400*x^2 + 9600*x^3 + 7200*x^4 + 2304*x^5 + 256*x^6)*exp(4*x)/45. - G. C. Greubel, Jul 21 2019
From Amiram Eldar, Mar 25 2022: (Start)
Sum_{n>=0} 1/a(n) = 8394/5 - 5832*log(4/3).
Sum_{n>=0} (-1)^n/a(n) = 75000*log(5/4) - 83674/5. (End)
MAPLE
seq(seq(binomial(i, j)*4^(i-6), j =i-6), i=6..36); # Zerinvary Lajos, Dec 03 2007
seq(binomial(n+6, 6)*4^n, n=0..30); # Zerinvary Lajos, Jun 16 2008
MATHEMATICA
Table[4^n*Binomial[n+6, 6], {n, 0, 30}] (* G. C. Greubel, Jul 21 2019 *)
PROG
(Sage) [lucas_number2(n, 4, 0)*binomial(n, 6)/2^12 for n in range(6, 36)] # Zerinvary Lajos, Mar 11 2009
(Magma) [4^n*Binomial(n+6, 6): n in [0..30]]; // Vincenzo Librandi, Oct 15 2011
(PARI) vector(30, n, n--; 4^n*binomial(n+6, 6) ) \\ G. C. Greubel, Jul 21 2019
(GAP) List([0..30], n-> 4^n*Binomial(n+6, 6)); # G. C. Greubel, Jul 21 2019
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Wolfdieter Lang, Mar 13 2000
STATUS
approved