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A013976 Number of tournaments on n nodes with a unique winner.

%I A013976 #20 Sep 09 2022 22:27:19
%S A013976 1,2,6,32,600,20544,1218224,160241152,42129744768,21293228876800,
%T A013976 22220602090444032,45959959305969143808,188702851949391611599872,
%U A013976 1566370829898884830446395392,25801614282517307546368494213120,851300764257208779000509484701188096
%N A013976 Number of tournaments on n nodes with a unique winner.
%H A013976 Andrew Howroyd, <a href="/A013976/b013976.txt">Table of n, a(n) for n = 1..20</a>
%H A013976 Yaakov Malinovsky and John W. Moon, <a href="https://arxiv.org/abs/2208.14932">On Round-Robin Tournaments with a Unique Maximum Score and Some Related Results</a>, arXiv:2208.14932 [math.CO], 2022.
%H A013976 <a href="/index/To#tournament">Index entries for sequences related to tournaments</a>
%o A013976 (PARI)
%o A013976 Winners(n)={
%o A013976   local(M=Map(Mat([x^0, 1])));
%o A013976   my(acc(p, v)=my(z); mapput(M, p, if(mapisdefined(M, p, &z), z+v, v)));
%o A013976   my(merge(r, p, v)=acc(p + sum(i=1, poldegree(p)-r-1, polcoef(p,i)*(1-x^i)), v));
%o A013976   my(recurse(r, p, i, q, v, e)=if(i<0, merge(r, x^e+q, v), my(t=polcoef(p, i)); for(k=0, t, self()(r, p, i-1, (k+x*(t-k))*x^i+q, binomial(t, k)*v, e+k))));
%o A013976   for(k=2, n, my(src=Mat(M)); M=Map(); for(i=1, matsize(src)[1], my(p=src[i, 1]); recurse(n-k, p, poldegree(p), 0, src[i, 2], 0)));
%o A013976   Mat(M);
%o A013976 }
%o A013976 a(n)={my(M=Winners(n)); sum(i=1, matsize(M)[1], if(pollead(M[i,1])==1, M[i,2]))} \\ _Andrew Howroyd_, Feb 29 2020
%Y A013976 Cf. A125031.
%K A013976 nonn,nice
%O A013976 1,2
%A A013976 Michael Stob (stob(AT)udu.calvin.edu)
%E A013976 a(1)=1 inserted and a(12) and beyond from _Andrew Howroyd_, Feb 28 2020