A089041 Inverse binomial transform of squares of factorial numbers.

1, 0, 3, 26, 453, 11844, 439975, 22056222, 1436236809, 117923229512, 11921584264011, 1455483251191650, 211163237294447053, 35913642489947449356, 7077505637217289437423, 1599980633296779087784934, 411293643476907595937924625, 119299057697083019137937718672

Offset

0,3

Comments

a(n) enumerates (ordered) lists of n two-tuples such that all numbers from 1 to n appear as the first as well as the second tuple entry and the j-th list member is not the tuple (j,j), for every j=1,..,n. Called coincidence-free 2-tuple lists of length n. See the Charalambides reference for this combinatorial interpretation.

References

Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 187, Exercise 13.(a), for r=2.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..200

Roland Bacher, Counting Packings of Generic Subsets in Finite Groups, Electr. J. Combinatorics, 19 (2012), #P7. - From N. J. A. Sloane, Feb 06 2013

Formula

G.f.: hypergeom([1, 1, 1], [], x/(1+x))/(1+x).

E.g.f.: exp(-x)* hypergeom([1, 1], [], x).

a(n) = n^2*a(n-1) + n*(n-1)*a(n-2) + (-1)^n. - Vladeta Jovovic, Jul 15 2004

a(n) = Sum_{j=0..n} ((-1)^(n-j))*binomial(n,j)*(j!)^2. See the Charalambides reference a(n)=B_{n,2}.

a(n) = (n-1)*(n+1)*a(n-1) + (n-1)*(2*n-1)*a(n-2) + (n-2)*(n-1)*a(n-3). - Vaclav Kotesovec, Aug 13 2013

a(n) ~ 2*Pi*exp(-2*n)*n^(2*n+1). - Vaclav Kotesovec, Aug 13 2013

G.f.: Sum_{k>=0} (k!)^2*x^k/(1 + x)^(k+1). - Ilya Gutkovskiy, Apr 12 2019

Example

2-tuple combinatorics: a(1)=0 because the only list of 2-tuples with numbers 1 is [(1,1)] and this is a coincidence for j=1.
2-tuple combinatorics: the a(2)=3 coincidence free 2-tuple lists of length n=2 are [(1,2),(2,1)], [(2,1),(1,2)] and [(2,2),(1,1)]. The list [(1,1),(2,2)] has two coincidences (j=1 and j=2).

Maple

a:= proc(n) option remember;
       `if`(n<2, 1-n, n^2*a(n-1)+n*(n-1)*a(n-2)+(-1)^n)
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Jun 21 2013

Mathematica

Table[n!Sum[(-1)^k(n-k)!/k!, {k, 0, n}], {n, 0, 15}] (* Geoffrey Critzer, Jun 17 2013 *)

Crossrefs

Cf. A001044, A046662 (binomial transform of squares of factorial numbers).

(-1)^n times the polynomials in A099599 evaluated at -1.

Sequence in context: A321183 A274778 A049088 * A059511 A112676 A103112

Adjacent sequences: A089038 A089039 A089040 * A089042 A089043 A089044

Keyword

easy,nonn

Author

Vladeta Jovovic, Dec 03 2003

Extensions

Charalambides reference and comments with combinatorial examples from Wolfdieter Lang, Jan 21 2008

Status

approved