%I #44 Jan 06 2025 06:31:12
%S 24,3648,90384,1304576,19958400,311592960,5054353920,85709352960,
%T 1523221539840,28387834675200,554575551129600,11345938174771200,
%U 242796629621145600,5427273747293798400,126546947417899008000
%N Number of n-node tournaments that have exactly five circular triads.
%H Ian R. Harris and Ryan P. A. McShane, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL27/McShane/mcshane1.html">Counting Tournaments with a Specified Number of Circular Triads</a>, Journal of Integer Sequences, Vol. 27 (2024), Article 24.8.7. See pages 2, 23.
%H J. B. Kadane, <a href="https://doi.org/10.1214/aoms/1177699532">Some equivalence classes in paired comparisons</a>, The Annals of Mathematical Statistics, 37 (1966), 488-494.
%F Kadane proves that a(n) = n!*((1/5)*(n-4)+(14/3)*(n-5)+8*(n-6)I(n>5)+(7/9)*(n-6)*(n-7)I(n>5)+(10/3)*(n-7)*(n-8)I(n>6)+(5/18)*(n-8)*(n-9)*(n-10)I(n>7)+(1/162)*(n-9)*(n-10)*(n-11)*(n-12)I(n>8)+(1/29160)*(n-10)*(n-11)*(n-12)*(n-13)*(n-14)I(n>9)), where I(p) is the indicator function: 1 if p is true and 0 otherwise.
%F E.g.f.: (5*x^10-180*x^9+2205*x^8-12150*x^7+34155*x^6-51840*x^5+38313*x^4-3942*x^3-11502*x^2+4698*x+243)*x^5/(5*3^5*(1-x)^6).
%Y Cf. A357242, A357248, A357257.
%K nonn,easy
%O 5,1
%A _Ian R Harris_, _Ryan P. A. McShane_, Sep 22 2022