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A052571 E.g.f. x^3/(1-x)^2.

%I A052571 #50 Feb 27 2025 17:24:02
%S A052571 0,0,0,6,48,360,2880,25200,241920,2540160,29030400,359251200,
%T A052571 4790016000,68497228800,1046139494400,16999766784000,292919058432000,
%U A052571 5335311421440000,102437979291648000,2067966706950144000
%N A052571 E.g.f. x^3/(1-x)^2.
%C A052571 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 A052571 Vincenzo Librandi, <a href="/A052571/b052571.txt">Table of n, a(n) for n = 0..300</a>
%H A052571 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Enumerative Formulas for Some Functions on Finite Sets</a>.
%H A052571 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=514">Encyclopedia of Combinatorial Structures 514</a>.
%H A052571 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 A052571 <a href="/index/Fa#facbase">Index entries for sequences related to factorial base representation</a>
%F A052571 E.g.f.: x^3/(-1+x)^2.
%F A052571 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 A052571 For n >= 2, a(n) = (n-2)*n!.
%F A052571 a(n+2) = n*(n+1)*(n+2)*n!. - _Zerinvary Lajos_, Nov 25 2006
%F A052571 a(n) = 3*A090672(n-2) = 6*A005990(n-2). - _Zerinvary Lajos_, May 11 2007
%F A052571 From _Amiram Eldar_, Jan 14 2021: (Start)
%F A052571 Sum_{n>=3} 1/a(n) = 9/4 - e - gamma/2 + Ei(1)/2 = 9/4 - A001113 - (1/2)*A001620 + (1/2)*A091725.
%F A052571 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 A052571 spec := [S,{S=Prod(Z,Z,Z,Sequence(Z),Sequence(Z))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%p A052571 [seq (n*(n+1)*(n+2)*n!,n=0..17)]; # _Zerinvary Lajos_, Nov 25 2006
%p A052571 a:=n->add((n!),j=1..n-2):seq(a(n), n=0..21); # _Zerinvary Lajos_, Aug 27 2008
%p A052571 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 A052571 Table[Sum[n!, {i, 3, n}], {n, 0, 19}] (* _Zerinvary Lajos_, Jul 12 2009 *)
%t A052571 With[{nn=20},CoefficientList[Series[x^3/(1-x)^2,{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Feb 27 2025 *)
%o A052571 (Magma) [0,0] cat [n*(n+1)*(n+2)*Factorial(n): n in [0..20]]; // _Vincenzo Librandi_, Oct 11 2011
%o A052571 (Scheme) (define (A052571 n) (if (< n 2) 0 (* (- n 2) (A000142 n)))) ;; _Antti Karttunen_, May 07 2015
%Y A052571 Column 5 of A257503 (apart from zero terms. Equally, row 5 of A257505).
%Y A052571 Cf. A000142, A007623, A005990, A090672.
%Y A052571 Cf. sequences with formula (n + k)*n! listed in A282466.
%Y A052571 Cf. A001113, A001620, A091725, A099285.
%K A052571 easy,nonn
%O A052571 0,4
%A A052571 encyclopedia(AT)pommard.inria.fr, Jan 25 2000