A006157 a(n+1) = (n-1)*a(n) + n*n!.

1, 5, 28, 180, 1320, 10920, 100800, 1028160, 11491200, 139708800, 1836172800, 25945920000, 392302310400, 6320426112000, 108101081088000, 1956280854528000, 37347179950080000, 750144785854464000, 15813863053148160000, 349121438173347840000

Offset: 2

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• nonn
• easy

Comments

Number of ascending runs of length at least two in all permutations of [n]. Example: a(3)=5 because we have (123), (13)2, 3(12), 2(13), (23)1 and 321, where the ascending runs of length at least 2 are shown between parentheses. - Emeric Deutsch and Ira M. Gessel, Sep 07 2004

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• G. C. Greubel, Table of n, a(n) for n = 2..445
• J. Francon, Histoires de fichiers, RAIRO Informatique Théorique et Applications, 12 (1978), 49-62.
• J. Francon, Histoires de fichiers, RAIRO Informatique Théorique et Applications, 12 (1978), 49-62. (Annotated scanned copy)

Crossrefs

Cf. A014484.

Programs

Magma

[(2*n-1)*Factorial(n)/6: n in [2..40]]; // G. C. Greubel, Jan 08 2025

Mathematica

Table[(2n-1)/6*n!,{n,2,30}] (* Harvey P. Dale, Jan 06 2014 *)

Python

def A006157(n): return (2*n-1)*factorial(n)//6
print([A006157(n) for n in range(2,41)]) # G. C. Greubel, Jan 08 2025

Formula

a(n) = (2n-1)/6 * n!.

E.g.f.: x^2*(3-x)/(6*(1-x)^2). - Emeric Deutsch and Ira M. Gessel, Sep 07 2004

Extensions

More terms from Harvey P. Dale, Jan 06 2014