A052520 Number of pairs of sequences of cardinality at least 2.

0, 0, 0, 0, 24, 240, 2160, 20160, 201600, 2177280, 25401600, 319334400, 4311014400, 62270208000, 958961203200, 15692092416000, 271996268544000, 4979623993344000, 96035605585920000, 1946321606541312000, 41359334139002880000, 919636959090769920000, 21356013827774545920000

Offset: 0

Author

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

Keywords

• easy
• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..400
• Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 87.

Crossrefs

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

Cf. A001113, A001620, A068985, A091725, A099285.

Programs

GAP

Concatenation([0,0,0,0], List([4..20], n-> (n-3)*Factorial(n))); # G. C. Greubel, May 13 2019

Magma

[n le 3 select 0 else (n-3)*Factorial(n): n in [0..20]]; // G. C. Greubel, May 13 2019

Maple

spec := [S,{B=Sequence(Z,2 <= card),S=Prod(B,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

Mathematica

Table[Sum[n!, {i,4,n}], {n, 0, 19}] (* Zerinvary Lajos, Jul 12 2009 *)
With[{nn=20},CoefficientList[Series[x^4/(x-1)^2,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jun 03 2016 *)

PARI

{a(n) = if(n<4, 0, (n-3)*n!)}; \\ G. C. Greubel, May 13 2019

Sage

[0,0,0,0]+[(n-3)*factorial(n) for n in (4..20)] # G. C. Greubel, May 13 2019

Formula

E.g.f.: x^4/(1-x)^2.

(n-3)*a(n+1) + (2+n-n^2)*a(n) = 0, with a(0) = a(1) = a(2) = a(3) = 0, a(4) = 24.

a(n) = (n-3)*n!, n>2.

a(n) = (n+1)!*(n-3)/(n+1), n>2. - Gary Detlefs, Oct 02 2011

From Amiram Eldar, Jan 14 2021: (Start)

Sum_{n>=4} 1/a(n) = 59/36 - 2*e/3 - gamma/6 + Ei(1)/6 = 59/36 - (2/3)*A001113 - (1/6)*A001620 + A091725/2.

Sum_{n>=4} (-1)^n/a(n) = 1/36 - 1/(3*e) + gamma/6 - Ei(-1)/6 = 1/36 - (1/3)*A068985 + (1/6)*A001620 + (1/6)*A099285. (End)