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%I #40 Sep 24 2022 05:47:38
%S 0,0,0,27,108,270,540,945,1512,2268,3240,4455,5940,7722,9828,12285,
%T 15120,18360,22032,26163,30780,35910,41580,47817,54648,62100,70200,
%U 78975,88452,98658,109620,121365,133920,147312,161568,176715,192780,209790,227772,246753
%N a(n) = (9/2)*(n-1)*(n-2)*(n-3).
%C 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
%H G. C. Greubel, <a href="/A134171/b134171.txt">Table of n, a(n) for n = 1..1000</a>
%H D. Zvonkine, <a href="http://www.math.jussieu.fr/~zvonkine/">Home Page</a>.
%H D. Zvonkine, <a href="http://mi.mathnet.ru/eng/mmj274">Counting ramified coverings and intersection theory on Hurwitz spaces II (local structure of Hurwitz spaces and combinatorial results)</a>, Moscow Mathematical Journal, Vol. 7, No. 1 (2007), 135-162.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F a(n) = 27 * binomial(n-1,3). - _Zerinvary Lajos_, Aug 06 2008
%F From _Chai Wah Wu_, May 29 2016: (Start)
%F a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>4.
%F G.f.: 27*x^4/(1-x)^4. (End)
%F E.g.f.: 27 + (9/2*(x^3-3*x^2+6*x-6))*exp(x). - _G. C. Greubel_, May 17 2021
%F a(n) = 27 * A000292(n-3) for n >= 3. - _Alois P. Heinz_, May 17 2021
%F From _Amiram Eldar_, Sep 24 2022: (Start)
%F Sum_{n>=4} 1/a(n) = 1/18.
%F Sum_{n>=4} (-1)^n/a(n) = 4*log(2)/9 - 5/18. (End)
%p seq(27*binomial(n-1, 3), n=1..30); # _Zerinvary Lajos_, May 18 2008
%t LinearRecurrence[{4,-6,4,-1}, {0,0,0,27}, 50] (* _G. C. Greubel_, May 29 2016 *)
%o (Magma) [(9/2)*(n-1)*(n-2)*(n-3) : n in [1..50]]; // _Wesley Ivan Hurt_, May 29 2016
%Y Cf. A008585, A027465, A027468. - _Zerinvary Lajos_, Aug 06 2008
%Y Cf. A000292, A102741, A113335.
%K nonn,easy
%O 1,4
%A _N. J. A. Sloane_, Jan 30 2008
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