OFFSET
1,4
COMMENTS
Number of n permutations (n>=3) of 4 objects u, v, z, x with repetition allowed, containing n-3=0 u's. Example: if n=3 then n-3 =zero u, a()=27 because we have vzx, vxz, zvx, zxv, xvz, xzv, vvv, zzz, xxx, vvx, vxv, xvv, xxv, xvx, vxx, vvz, vzv, zvv, zzv, zvz, vzz, xzz, zxz, zzx, xxz, xzx, zxx. A027465 formatted as a triangular array: diagonal: 27, 108, 270, 540, 945, 1512. - Zerinvary Lajos, Aug 06 2008
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
D. Zvonkine, Home Page.
D. Zvonkine, Counting ramified coverings and intersection theory on Hurwitz spaces II (local structure of Hurwitz spaces and combinatorial results), Moscow Mathematical Journal, Vol. 7, No. 1 (2007), 135-162.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).
FORMULA
a(n) = 27 * binomial(n-1,3). - Zerinvary Lajos, Aug 06 2008
From Chai Wah Wu, May 29 2016: (Start)
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>4.
G.f.: 27*x^4/(1-x)^4. (End)
E.g.f.: 27 + (9/2*(x^3-3*x^2+6*x-6))*exp(x). - G. C. Greubel, May 17 2021
a(n) = 27 * A000292(n-3) for n >= 3. - Alois P. Heinz, May 17 2021
From Amiram Eldar, Sep 24 2022: (Start)
Sum_{n>=4} 1/a(n) = 1/18.
Sum_{n>=4} (-1)^n/a(n) = 4*log(2)/9 - 5/18. (End)
MAPLE
seq(27*binomial(n-1, 3), n=1..30); # Zerinvary Lajos, May 18 2008
MATHEMATICA
LinearRecurrence[{4, -6, 4, -1}, {0, 0, 0, 27}, 50] (* G. C. Greubel, May 29 2016 *)
PROG
(Magma) [(9/2)*(n-1)*(n-2)*(n-3) : n in [1..50]]; // Wesley Ivan Hurt, May 29 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Jan 30 2008
STATUS
approved