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A360820
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a(n) = Sum_{k=0..n} binomial(n, k)*2^(n^2-k*(n-k)).
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0
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1, 4, 48, 1792, 221184, 98566144, 173946175488, 1281755680079872, 39534286378918477824, 5018464395368794081460224, 2586745980900067184722499862528, 5375203895735606878055792019528220672, 44865714160227204455469409035569750630989824, 1501355804811017489524770237231795462175548447391744
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OFFSET
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0,2
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COMMENTS
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This is the model count of the following sentence in first-order logic: forall X. forall Y. friends(X, Y) /\ smokes(X) -> smokes(Y).
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LINKS
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FORMULA
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a(n) = Sum_{k=0..n} binomial(n, k)*2^(n^2-k*(n-k)).
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EXAMPLE
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If the domain is {1}, then all four interpretations ({}, {smokes(1)}, {friends(1, 1)}, {smokes(1), friends(1, 1)}) are models, so a(1) = 4.
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MATHEMATICA
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a[n_] := Sum[Binomial[n, k]*2^(n^2 - k*(n - k)), {k, 0, n}]; Array[a, 14, 0] (* Amiram Eldar, Feb 24 2023 *)
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PROG
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(PARI) a(n) = sum(k=0, n, binomial(n, k)*2^(n^2-k*(n-k))); \\ Michel Marcus, Feb 22 2023
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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