A134171 a(n) = (9/2)*(n-1)*(n-2)*(n-3).

0, 0, 0, 27, 108, 270, 540, 945, 1512, 2268, 3240, 4455, 5940, 7722, 9828, 12285, 15120, 18360, 22032, 26163, 30780, 35910, 41580, 47817, 54648, 62100, 70200, 78975, 88452, 98658, 109620, 121365, 133920, 147312, 161568, 176715, 192780, 209790, 227772, 246753

Offset: 1

Author

N. J. A. Sloane, Jan 30 2008

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A008585, A027465, A027468. - Zerinvary Lajos, Aug 06 2008

Cf. A000292, A102741, A113335.

Programs

Magma

[(9/2)*(n-1)*(n-2)*(n-3) : n in [1..50]]; // Wesley Ivan Hurt, May 29 2016

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 *)

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)