A038048 a(n) = (n-1)! * sigma(n).

1, 3, 8, 42, 144, 1440, 5760, 75600, 524160, 6531840, 43545600, 1117670400, 6706022400, 149448499200, 2092278988800, 40537905408000, 376610217984000, 13871809695744000, 128047474114560000, 5109094217170944000

Offset: 1

Author

Christian G. Bower

Keywords

• nonn
• nice
• easy

Comments

sigma(n) = A000203(n) is the sum of the divisors of n.

Number of labeled regular octopi (or octopuses, cycles of ordered sets all the same size).

Left edge of triangle in A008298.

Examples

			a(6) = 5! * (1 + 2 + 3 + 6) = 1440 = 6! * (1 + 1/2 + 1/3 + 1/6).
		

References

• F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 56 (1.4.67).
• L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 159, #10, A(n,1).

Links

• T. D. Noe, Table of n, a(n) for n = 1..100
• Xiaojun Liu, Motohico Mulase, Adam Sorkin, Quantum curves for simple Hurwitz numbers of an arbitrary base curve, arXiv:1304.0015 [math.AG], 2013.
• H. Ochiai, Counting functions for branched covers of elliptic curves and quasi-modular forms, arXiv:math-ph/9909023, 1999.

Crossrefs

Cf. A000203, A057625, A058892, A110373, A110374.

Programs

Maple

a := n -> n!*add(1/j, j=numtheory:-divisors(n)): seq(a(n), n=1..23); # Emeric Deutsch, Jul 24 2005

Mathematica

a[n_] := (n-1)!*DivisorSigma[1, n]; Table[a[n], {n, 20}] (* Jean-François Alcover, Mar 23 2011 *)

PARI

a(n)=(n-1)!*sigma(n) \\ Charles R Greathouse IV, Mar 09 2014

Sage

A038048 = lambda n: factorial(n-1)*sigma(n,1)
[A038048(n) for n in (1..20)] # Peter Luschny, Jan 19 2016

Formula

a(n) = Sum_{d|n} n!/d. - Amarnath Murthy, Jul 24 2005

a(p) = (p+1)*(p-1)! if p is a prime. - Amarnath Murthy, Jul 24 2005

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

E.g.f.: Sum_{k>0} x^k/(k*(1-x^k)). - Vladeta Jovovic, Mar 27 2005

a(n) = A000142(n-1)*A000203(n). - Omar E. Pol, Feb 26 2014

Extensions

More terms from Emeric Deutsch, Jul 24 2005

Edited by N. J. A. Sloane, May 12 2008 at the suggestion of Joerg Arndt