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(PARI) a(n) = if (n == -1, 7, sum(k=0, 3, (n+k)!*binomial(3, k))); \\ _Michel Marcus_, Jun 30 2017
a(n) = Sum _{k = 0..3} (n+k)! C(3,k), k = 0..3.
Biondi, E.; Divieti, L.; Guardabassi, G.; Counting paths, circuits, chains and cycles in graphs: A unified approach. Canad. J. Math. 22 1970 22-35.
(PARI) a(n) = if (n == -1, 7, sum(k=0, 3, (n+k)!*binomial(3, k)));
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