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%I #54 Mar 02 2019 23:33:09
%S 1,3,8,42,144,1440,5760,75600,524160,6531840,43545600,1117670400,
%T 6706022400,149448499200,2092278988800,40537905408000,376610217984000,
%U 13871809695744000,128047474114560000,5109094217170944000
%N a(n) = (n-1)! * sigma(n).
%C sigma(n) = A000203(n) is the sum of the divisors of n.
%C Number of labeled regular octopi (or octopuses, cycles of ordered sets all the same size).
%C Left edge of triangle in A008298.
%D F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 56 (1.4.67).
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 159, #10, A(n,1).
%H T. D. Noe, <a href="/A038048/b038048.txt">Table of n, a(n) for n = 1..100</a>
%H Xiaojun Liu, Motohico Mulase, Adam Sorkin, <a href="http://arxiv.org/abs/1304.0015">Quantum curves for simple Hurwitz numbers of an arbitrary base curve</a>, arXiv:1304.0015 [math.AG], 2013.
%H H. Ochiai, <a href="http://arXiv.org/abs/math-ph/9909023">Counting functions for branched covers of elliptic curves and quasi-modular forms</a>, arXiv:math-ph/9909023, 1999.
%F a(n) = Sum_{d|n} n!/d. - _Amarnath Murthy_, Jul 24 2005
%F a(p) = (p+1)*(p-1)! if p is a prime. - _Amarnath Murthy_, Jul 24 2005
%F E.g.f.: log(f(x)), where f(x) = o.g.f. for partitions (A000041), Product_{k>=1} 1/(1 - x^k). - _N. J. A. Sloane_
%F E.g.f.: Sum_{k>0} x^k/(k*(1-x^k)). - _Vladeta Jovovic_, Mar 27 2005
%F a(n) = A000142(n-1)*A000203(n). - _Omar E. Pol_, Feb 26 2014
%e a(6) = 5! * (1 + 2 + 3 + 6) = 1440 = 6! * (1 + 1/2 + 1/3 + 1/6).
%p a := n -> n!*add(1/j, j=numtheory:-divisors(n)): seq(a(n), n=1..23); # _Emeric Deutsch_, Jul 24 2005
%t a[n_] := (n-1)!*DivisorSigma[1, n]; Table[a[n], {n, 20}] (* _Jean-François Alcover_, Mar 23 2011 *)
%o (PARI) a(n)=(n-1)!*sigma(n) \\ _Charles R Greathouse IV_, Mar 09 2014
%o (Sage)
%o A038048 = lambda n: factorial(n-1)*sigma(n,1)
%o [A038048(n) for n in (1..20)] # _Peter Luschny_, Jan 19 2016
%Y Cf. A000203, A057625, A058892, A110373, A110374.
%K nonn,nice,easy
%O 1,2
%A _Christian G. Bower_
%E More terms from _Emeric Deutsch_, Jul 24 2005
%E Edited by _N. J. A. Sloane_, May 12 2008 at the suggestion of _Joerg Arndt_
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