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Number of labeled split graphs on n vertices such that {1..k} is independent and {k+1..n} is a clique for some k in {0..n}.
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%I #33 May 28 2024 16:36:32

%S 1,2,6,24,128,928,9280,129152,2515200,68780544,2647000064,

%T 143580989440,10988411686912,1187350176604160,181232621966082048,

%U 39089521693818912768,11916533065969825808384,5135497592471003032846336,3128995097443083790244380672,2695613904312277811648715554816

%N Number of labeled split graphs on n vertices such that {1..k} is independent and {k+1..n} is a clique for some k in {0..n}.

%C Also the number of sign mappings X:([n] choose 2) -> {+,-} such that for any ordered 3-tuple abc we have X(ab)X(ac)X(bc) not in {++-,+--}.

%F a(n) = 1 + Sum_{k=1..n-1} (2^k-1)*2^((n-1-k)*k).

%p seq(1 + add((2^k-1)*2^((n-1-k)*k),k=1..n-1),n=1..20); # Georg Fischer_, May 28 2024

%o (Python) def f(n): return 1+sum((2**k-1)*2**((n-1-k)*k) for k in range(1,n))

%Y Cf. A048194.

%K nonn

%O 1,2

%A _Robert Lauff_ and _Manfred Scheucher_, Jan 05 2024

%E a(20), a(21) joined by _Georg Fischer_, May 28 2024