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A089041 Inverse binomial transform of squares of factorial numbers. 4
1, 0, 3, 26, 453, 11844, 439975, 22056222, 1436236809, 117923229512, 11921584264011, 1455483251191650, 211163237294447053, 35913642489947449356, 7077505637217289437423, 1599980633296779087784934, 411293643476907595937924625, 119299057697083019137937718672 (list; graph; refs; listen; history; text; internal format)
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

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Last modified September 19 18:53 EDT 2020. Contains 337181 sequences. (Running on oeis4.)