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

Author

Vladeta Jovovic, Dec 03 2003

Keywords

• easy
• nonn

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.

Examples

			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).
		

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

Crossrefs

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

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

Programs

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 *)

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

Extensions

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