A052521 Number of pairs of sequences of cardinality at least 3.

0, 0, 0, 0, 0, 0, 720, 10080, 120960, 1451520, 18144000, 239500800, 3353011200, 49816166400, 784604620800, 13076743680000, 230150688768000, 4268249137152000, 83230858174464000, 1703031405723648000

Offset: 0

Author

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

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..445
• Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 88.

Crossrefs

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

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

Programs

GAP

Concatenation([0,0,0,0,0,0], List([6..20], n-> (n-5)*Factorial(n))); # G. C. Greubel, May 13 2019

Magma

[n le 5 select 0 else (n-5)*Factorial(n): n in [0..20]]; // G. C. Greubel, May 13 2019

Maple

spec := [S,{B=Sequence(Z,3 <= card), S=Prod(B,B)},labeled]: # Pairs spec
seq(combstruct[count](spec, size=n), n=0..20);

Mathematica

Table[If[n<6, 0, (n-5)*n!], {n,0,20}] (* G. C. Greubel, May 13 2019 *)

PARI

{a(n) = if(n<6, 0, (n-5)*n!)}; \\ G. C. Greubel, May 13 2019

Sage

[0,0,0,0,0,0]+[(n-5)*factorial(n) for n in (6..20)] # G. C. Greubel, May 13 2019

Formula

E.g.f.: x^6/(1-x)^2.

(n-5)*a(n+1) + (4 + 3*n - n^2)*a(n) = 0, with a(0) = a(1) = a(2) = a(3) = a(4) = a(5) = 0, a(6) = 720.

a(n) = (n-5)*n!.

From Amiram Eldar, Jan 14 2021: (Start)

Sum_{n>=6} 1/a(n) = 5477/7200 - 17*e/60 - gamma/120 + Ei(1)/120 = 5477/7200 - (17/60)*A001113 - (1/120)*A001620 + A091725/120.

Sum_{n>=6} (-1)^n/a(n) = 403/7200 - 1/(6*e) + gamma/120 - Ei(-1)/120 = 403/7200 - (1/6)*A068985 + (1/120)*A001620 + (1/120)*A099285. (End)