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

0, 0, 0, 6, 48, 360, 2880, 25200, 241920, 2540160, 29030400, 359251200, 4790016000, 68497228800, 1046139494400, 16999766784000, 292919058432000, 5335311421440000, 102437979291648000, 2067966706950144000

Offset: 0

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

• easy
• nonn

Comments

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..300
• Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 514.
• Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint, arXiv:1406.3081 [math.CO], 2014-2015.
• Index entries for sequences related to factorial base representation

Crossrefs

Column 5 of A257503 (apart from zero terms. Equally, row 5 of A257505).

Cf. A000142, A007623, A005990, A090672.

Cf. sequences with formula (n + k)*n! listed in A282466.

Cf. A001113, A001620, A091725, A099285.

Programs

Magma

[0,0] cat [n*(n+1)*(n+2)*Factorial(n): n in [0..20]]; // Vincenzo Librandi, Oct 11 2011

Maple

spec := [S,{S=Prod(Z,Z,Z,Sequence(Z),Sequence(Z))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
[seq (n*(n+1)*(n+2)*n!,n=0..17)]; # Zerinvary Lajos, Nov 25 2006
a:=n->add((n!),j=1..n-2):seq(a(n), n=0..21); # Zerinvary Lajos, Aug 27 2008
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

Mathematica

Table[Sum[n!, {i, 3, n}], {n, 0, 19}] (* Zerinvary Lajos, Jul 12 2009 *)
With[{nn=20},CoefficientList[Series[x^3/(1-x)^2,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Feb 27 2025 *)

Scheme

(define (A052571 n) (if (< n 2) 0 (* (- n 2) (A000142 n)))) ;; Antti Karttunen, May 07 2015

Formula

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

Recurrence: {a(1)=0, a(0)=0, a(2)=0, a(3)=6, (1-n^2)*a(n)+(-2+n)*a(n+1)=0}.

For n >= 2, a(n) = (n-2)*n!.

a(n+2) = n*(n+1)*(n+2)*n!. - Zerinvary Lajos, Nov 25 2006

a(n) = 3*A090672(n-2) = 6*A005990(n-2). - Zerinvary Lajos, May 11 2007

From Amiram Eldar, Jan 14 2021: (Start)

Sum_{n>=3} 1/a(n) = 9/4 - e - gamma/2 + Ei(1)/2 = 9/4 - A001113 - (1/2)*A001620 + (1/2)*A091725.

Sum_{n>=3} (-1)^(n+1)/a(n) = -1/4 + gamma/2 - Ei(-1)/2 = -1/4 + (1/2)*A001620 + (1/2)*A099285. (End)