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Number of homogeneous games for n players.
2

%I #37 Aug 24 2023 05:15:40

%S 0,1,3,8,23,76,293,1307,6642,37882

%N Number of homogeneous games for n players.

%H I. Krohn and P. Sudhölter, <a href="https://doi.org/10.1007/BF01415753">Directed and weighted majority games</a>, Mathematical Methods of Operation Research, 42, 2 (1995), 189-216. See Table 1, p. 213; also on <a href="https://www.researchgate.net/publication/226788682_Directed_and_weighted_majority_games">ResearchGate</a>.

%H P. Sudhölter, <a href="https://doi.org/10.1007/BF01358803">Homogeneous games as anti step functions</a>, International Journal of Game Theory Vol. 18, (1989), 433-469.

%F Conjecture: g.f.: Q(0)*x/(1-x), where Q(k) = 1 + (1-(1-x)^(2*k+2))/(1- (1-(1-x)^(2*k+3))/(1-(1-x)^(2*k+3) + 1/Q(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 13 2013

%F Note that a(n) - a(n-1) = A022493(n) for 1 <= n <= 9. Does this equality hold for n > 9? If so, then we have the g.f. 1/(1 - x)*( Sum_{n >= 1} Product_{k = 1..n} (1 - (1 - x)^k) ). - _Peter Bala_, Dec 13 2021

%Y Subclass of A000617. Cf. A001532, A022493, A109456, A132183.

%K nonn,hard,more

%O 0,3

%A _Fabián Riquelme_, Apr 20 2011