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A379614 a(n) = Sum_{k=0..n} 2^((n + k - 1)*(n - k)/2) * n! / (n - k)!. Row sums of A365638.

%I A379614 #17 Mar 19 2025 13:57:58
%S A379614 1,2,8,62,920,26104,1420496,148881328,30184791424,11884978702720,
%T A379614 9119462417850112,13676896785924429056,40193177909606893812736,
%U A379614 231954028321633491248270336,2633539132301114752459257620480,58919202504421532088999537276311552,2601059501207939343360641860628003520512
%N A379614 a(n) = Sum_{k=0..n} 2^((n + k - 1)*(n - k)/2) * n! / (n - k)!. Row sums of A365638.
%C A379614 Number of transitive tournaments that can occur as a subtournament of a tournament on n labeled vertices.
%H A379614 G. C. Greubel, <a href="/A379614/b379614.txt">Table of n, a(n) for n = 0..81</a>
%F A379614 a(n) ~ sqrt(Pi/log(2)) * 2^(n*(n-1)/2 + 5/8) * n^((1 + log(n)/log(2))/2). - _Vaclav Kotesovec_, Mar 19 2025
%p A379614 a := n -> add(2^(((n + k - 1)*(n - k))/2) * n! / (n - k)!, k = 0..n): seq(a(n), n = 0..16);
%t A379614 Table[Sum[2^(((n + k - 1)*(n - k))/2)*n!/(n - k)!, {k, 0, n} ], {n, 0, 16}] (* _Michael De Vlieger_, Dec 31 2024 *)
%o A379614 (Magma)
%o A379614 A379614:= func< n | (&+[Factorial(k)*Binomial(n,k)*2^(Binomial(n-k,2) +k*(n-k)): k in [0..n]]) >;
%o A379614 [A379614(n): n in [0..20]]; // _G. C. Greubel_, Mar 19 2025
%o A379614 (SageMath)
%o A379614 def A379614(n): return sum(factorial(k)*binomial(n,k)*2^(binomial(n-k,2) +k*(n-k)) for k in range(n+1))
%o A379614 print([A379614(n) for n in range(21)]) # _G. C. Greubel_, Mar 19 2025
%Y A379614 Cf. A365638, A382240.
%K A379614 nonn
%O A379614 0,2
%A A379614 _Peter Luschny_, Dec 31 2024