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a(n+1) = (n-1)*a(n) + n*n!.
(Formerly M3950)
8

%I M3950 #28 Nov 03 2017 22:18:11

%S 1,5,28,180,1320,10920,100800,1028160,11491200,139708800,1836172800,

%T 25945920000,392302310400,6320426112000,108101081088000,

%U 1956280854528000,37347179950080000,750144785854464000,15813863053148160000,349121438173347840000

%N a(n+1) = (n-1)*a(n) + n*n!.

%C 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

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

%H J. Francon, <a href="http://www.numdam.org/item?id=ITA_1978__12_1_49_0">Histoires de fichiers</a>, RAIRO Informatique Théorique et Applications, 12 (1978), 49-62.

%H J. Francon, <a href="/A006157/a006157.pdf">Histoires de fichiers</a>, RAIRO Informatique Théorique et Applications, 12 (1978), 49-62. (Annotated scanned copy)

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

%F E.g.f.: x^2*(3-x)/(6*(1-x)^2). - _Emeric Deutsch_ and _Ira M. Gessel_, Sep 07 2004

%t Table[(2n-1)/6*n!,{n,2,30}] (* _Harvey P. Dale_, Jan 06 2014 *)

%Y Cf. A014484.

%K nonn,easy

%O 2,2

%A _N. J. A. Sloane_, _Simon Plouffe_

%E More terms from _Harvey P. Dale_, Jan 06 2014