cp's OEIS Frontend

A089041 Inverse binomial transform of squares of factorial numbers.

%I A089041 #39 Apr 21 2025 13:24:29
%S A089041 1,0,3,26,453,11844,439975,22056222,1436236809,117923229512,
%T A089041 11921584264011,1455483251191650,211163237294447053,
%U A089041 35913642489947449356,7077505637217289437423,1599980633296779087784934,411293643476907595937924625,119299057697083019137937718672
%N A089041 Inverse binomial transform of squares of factorial numbers.
%C A089041 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.
%D A089041 Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 187, Exercise 13.(a), for r=2.
%H A089041 Alois P. Heinz, <a href="/A089041/b089041.txt">Table of n, a(n) for n = 0..200</a>
%H A089041 Roland Bacher, <a href="https://doi.org/10.37236/2522">Counting Packings of Generic Subsets in Finite Groups</a>, Electr. J. Combinatorics, 19 (2012), #P7. - From _N. J. A. Sloane_, Feb 06 2013
%F A089041 G.f.: hypergeom([1, 1, 1], [], x/(1+x))/(1+x).
%F A089041 E.g.f.: exp(-x)* hypergeom([1, 1], [], x).
%F A089041 a(n) = n^2*a(n-1) + n*(n-1)*a(n-2) + (-1)^n. - _Vladeta Jovovic_, Jul 15 2004
%F A089041 a(n) = Sum_{j=0..n} ((-1)^(n-j))*binomial(n,j)*(j!)^2. See the Charalambides reference a(n)=B_{n,2}.
%F A089041 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
%F A089041 a(n) ~ 2*Pi*exp(-2*n)*n^(2*n+1). - _Vaclav Kotesovec_, Aug 13 2013
%F A089041 G.f.: Sum_{k>=0} (k!)^2*x^k/(1 + x)^(k+1). - _Ilya Gutkovskiy_, Apr 12 2019
%e A089041 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.
%e A089041 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).
%p A089041 a:= proc(n) option remember;
%p A089041        `if`(n<2, 1-n, n^2*a(n-1)+n*(n-1)*a(n-2)+(-1)^n)
%p A089041     end:
%p A089041 seq(a(n), n=0..20);  # _Alois P. Heinz_, Jun 21 2013
%t A089041 Table[n!Sum[(-1)^k(n-k)!/k!,{k,0,n}],{n,0,15}] (* _Geoffrey Critzer_, Jun 17 2013 *)
%Y A089041 Cf. A001044, A046662 (binomial transform of squares of factorial numbers).
%Y A089041 (-1)^n times the polynomials in A099599 evaluated at -1.
%K A089041 easy,nonn
%O A089041 0,3
%A A089041 _Vladeta Jovovic_, Dec 03 2003
%E A089041 Charalambides reference and comments with combinatorial examples from _Wolfdieter Lang_, Jan 21 2008