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a(n) = 2*n^(n-2).
3

%I #12 Aug 01 2022 08:21:45

%S 2,2,6,32,250,2592,33614,524288,9565938,200000000,4715895382,

%T 123834728448,3584320788074,113387824750592,3892390136718750,

%U 144115188075855872,5724846103019631586,242879062193188503552

%N a(n) = 2*n^(n-2).

%C When n >=2, right side of binomial sum n-> Sum_{i=1..n-1} ( i^(n-i-1) * (n-i)^(i-1) *binomial(n, i) ). - Yong Kong (ykong(AT)curagen.com), Dec 28 2000

%D A. P. Prudnikov, Yu. A. Brychkov and O.I. Marichev, "Integrals and Series", Volume 1: "Elementary Functions", Chapter 4: "Finite Sums", New York, Gordon and Breach Science Publishers, 1986-1992, Eq. (4.2.2.38)

%H G. C. Greubel, <a href="/A003308/b003308.txt">Table of n, a(n) for n = 1..250</a>

%F a(n) = 2*n^(n-2).

%F a(n) = 2 * A000272(n).

%F E.g.f.: (-2)*Integral_{t=0..x} LambertW(-t)/t dt = (-1)*LambertW(-x) * (LambertW(-x) + 2). - _G. C. Greubel_, Jul 31 2022

%t Table[2*n^(n-2), {n,20}] (* _Harvey P. Dale_, Sep 18 2021 *)

%o (Magma) [2*n^(n-2): n in [1..30]]; // _G. C. Greubel_, Jul 31 2022

%o (SageMath) [2*n^(n-2) for n in (1..30)] # _G. C. Greubel_, Jul 31 2022

%Y Cf. A000272.

%K nonn

%O 1,1

%A Joseph Moser (jmoser(AT)wcupa.edu)