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%I #38 Sep 08 2022 08:45:28
%S 0,1,2,6,32,320,6144,229376,16777216,2415919104,687194767360,
%T 387028092977152,432345564227567616,959230691832896684032,
%U 4231240368651202111471616,37138201178561408246973726720,649037107316853453566312041152512,22596875928343569839364720024765857792
%N Total number of "Emperors" in all tournaments on n labeled nodes.
%C An "Emperor" is a player who beats everybody else.
%C a(n) is the number of isolated nodes in all simple labeled graphs on n nodes. - _Geoffrey Critzer_, Oct 19 2011
%H Alois P. Heinz, <a href="/A123903/b123903.txt">Table of n, a(n) for n = 0..50</a>
%H S. B. Maurer, <a href="http://www.maa.org/programs/maa-awards/writing-awards/the-king-chicken-theorems">The king chicken theorems</a>, Math. Mag., 53 (1980), 67-80.
%F a(n) = n*2^((n-1)*(n-2)/2).
%F E.g.f.: x * Sum_{n>=0} 2^C(n,2) x^n/n!. - _Geoffrey Critzer_, Oct 19 2011
%F a(n) = n * A006125(n-1). - _Anton Zakharov_, Dec 21 2016
%p a:= n-> n*2^((n-1)*(n-2)/2):
%p seq(a(n), n=0..20); # _Alois P. Heinz_, Aug 26 2013
%t a=Sum[2^Binomial[n,2]x^n/n!,{n,0,20}];
%t Range[0,20]!CoefficientList[Series[x a,{x,0,20}],x]
%t Table[n*2^Binomial[n-1,2], {n,0,20}] (* _G. C. Greubel_, Aug 06 2019 *)
%o (Maxima) A123903(n):=n*2^((n-1)*(n-2)/2)$ makelist(A123903(n),n,0,17); /* _Martin Ettl_, Nov 13 2012 */
%o (PARI) vector(20, n, n--; n*2^binomial(n-1,2)) \\ _G. C. Greubel_, Aug 06 2019
%o (Magma) [n*2^Binomial(n-1,2): n in [0..20]]; // _G. C. Greubel_, Aug 06 2019
%o (Sage) [n*2^binomial(n-1,2) for n in (0..20)] # _G. C. Greubel_, Aug 06 2019
%o (GAP) List([0..20], n-> n*2^Binomial(n-1,2)); # _G. C. Greubel_, Aug 06 2019
%Y Cf. A123553, A125031, A006125.
%K nonn
%O 0,3
%A _N. J. A. Sloane_, Nov 20 2006
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