A123903 Total number of "Emperors" in all tournaments on n labeled nodes.

0, 1, 2, 6, 32, 320, 6144, 229376, 16777216, 2415919104, 687194767360, 387028092977152, 432345564227567616, 959230691832896684032, 4231240368651202111471616, 37138201178561408246973726720, 649037107316853453566312041152512, 22596875928343569839364720024765857792

Offset: 0

Author

N. J. A. Sloane, Nov 20 2006

Keywords

• nonn

Comments

An "Emperor" is a player who beats everybody else.

a(n) is the number of isolated nodes in all simple labeled graphs on n nodes. - Geoffrey Critzer, Oct 19 2011

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..50
• S. B. Maurer, The king chicken theorems, Math. Mag., 53 (1980), 67-80.

Crossrefs

Cf. A123553, A125031, A006125.

Programs

GAP

List([0..20], n-> n*2^Binomial(n-1,2)); # G. C. Greubel, Aug 06 2019

Magma

[n*2^Binomial(n-1,2): n in [0..20]]; // G. C. Greubel, Aug 06 2019

Maple

a:= n-> n*2^((n-1)*(n-2)/2):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 26 2013

Mathematica

a=Sum[2^Binomial[n,2]x^n/n!,{n,0,20}];
Range[0,20]!CoefficientList[Series[x a,{x,0,20}],x]
Table[n*2^Binomial[n-1,2], {n,0,20}] (* G. C. Greubel, Aug 06 2019 *)

Maxima

A123903(n):=n*2^((n-1)*(n-2)/2)$ makelist(A123903(n),n,0,17); /* Martin Ettl, Nov 13 2012 */

PARI

vector(20, n, n--; n*2^binomial(n-1,2)) \\ G. C. Greubel, Aug 06 2019

Sage

[n*2^binomial(n-1,2) for n in (0..20)] # G. C. Greubel, Aug 06 2019

Formula

a(n) = n*2^((n-1)*(n-2)/2).

E.g.f.: x * Sum_{n>=0} 2^C(n,2) x^n/n!. - Geoffrey Critzer, Oct 19 2011

a(n) = n * A006125(n-1). - Anton Zakharov, Dec 21 2016