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%I #44 Sep 08 2022 08:44:59
%S 0,0,0,6,48,360,2880,25200,241920,2540160,29030400,359251200,
%T 4790016000,68497228800,1046139494400,16999766784000,292919058432000,
%U 5335311421440000,102437979291648000,2067966706950144000
%N E.g.f. x^3/(1-x)^2.
%C For n >= 3, a(n) = number whose factorial base representation (A007623) begins with digit {n-2} followed by n-1 zeros. Viewed in that base, this sequence looks like this: 0, 0, 0, 100, 2000, 30000, 400000, 5000000, 60000000, 700000000, 8000000000, 90000000000, A00000000000, B000000000000, ... (where "digits" A and B stand for placeholder values 10 and 11 respectively). - _Antti Karttunen_, May 07 2015
%H Vincenzo Librandi, <a href="/A052571/b052571.txt">Table of n, a(n) for n = 0..300</a>
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Enumerative Formulas for Some Functions on Finite Sets</a>.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=514">Encyclopedia of Combinatorial Structures 514</a>.
%H Luis Manuel Rivera, <a href="http://arxiv.org/abs/1406.3081">Integer sequences and k-commuting permutations</a>, arXiv preprint, arXiv:1406.3081 [math.CO], 2014-2015.
%H <a href="/index/Fa#facbase">Index entries for sequences related to factorial base representation</a>
%F E.g.f.: x^3/(-1+x)^2.
%F Recurrence: {a(1)=0, a(0)=0, a(2)=0, a(3)=6, (1-n^2)*a(n)+(-2+n)*a(n+1)=0}.
%F For n >= 2, a(n) = (n-2)*n!.
%F a(n+2) = n*(n+1)*(n+2)*n!. - _Zerinvary Lajos_, Nov 25 2006
%F a(n) = 3*A090672(n-2) = 6*A005990(n-2). - _Zerinvary Lajos_, May 11 2007
%F From _Amiram Eldar_, Jan 14 2021: (Start)
%F Sum_{n>=3} 1/a(n) = 9/4 - e - gamma/2 + Ei(1)/2 = 9/4 - A001113 - (1/2)*A001620 + (1/2)*A091725.
%F Sum_{n>=3} (-1)^(n+1)/a(n) = -1/4 + gamma/2 - Ei(-1)/2 = -1/4 + (1/2)*A001620 + (1/2)*A099285. (End)
%p spec := [S,{S=Prod(Z,Z,Z,Sequence(Z),Sequence(Z))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%p [seq (n*(n+1)*(n+2)*n!,n=0..17)]; # _Zerinvary Lajos_, Nov 25 2006
%p a:=n->add((n!),j=1..n-2):seq(a(n), n=0..21); # _Zerinvary Lajos_, Aug 27 2008
%p G(x):=x^3/(1-x)^2: f[0]:=G(x): for n from 1 to 21 do f[n]:=diff(f[n-1],x) od: x:=0: seq(f[n],n=0..19); # _Zerinvary Lajos_, Apr 01 2009
%t Table[Sum[n!, {i, 3, n}], {n, 0, 19}] (* _Zerinvary Lajos_, Jul 12 2009 *)
%o (Magma) [0,0],[n*(n+1)*(n+2)*Factorial(n): n in [0..20]]; // _Vincenzo Librandi_, Oct 11 2011
%o (Scheme) (define (A052571 n) (if (< n 2) 0 (* (- n 2) (A000142 n)))) ;; _Antti Karttunen_, May 07 2015
%Y Column 5 of A257503 (apart from zero terms. Equally, row 5 of A257505).
%Y Cf. A000142, A007623, A005990, A090672.
%Y Cf. sequences with formula (n + k)*n! listed in A282466.
%Y Cf. A001113, A001620, A091725, A099285.
%K easy,nonn
%O 0,4
%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000
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