%I #49 Feb 04 2024 16:36:11
%S 1,2,7,31,165,1031,7423,60621,554249,5611771,62353011,754471433,
%T 9876716941,139097096919,2097156230471,33704296561141,575219994643473,
%U 10389911153247731,198019483156015579,3971390745517868001,83608226221428800021,1843561388182505040463
%N First difference of A002627.
%C Previous name was: Q-residue of the triangle A094727, where Q is the triangular array (t(i,j)) given by t(i,j)=1. For the definition of Q-residue, see A193649.
%C Number of n X n rook placements avoiding the pattern 001. - _N. J. A. Sloane_, Feb 04 2013
%C Let M(n) denote the n X n matrix with ones along the subdiagonal, ones everywhere above the main diagonal, the integers 2, 3, etc., along the main diagonal, and zeros everywhere else. Then a(n) is equal to the permanent of M(n). - _John M. Campbell_, Apr 20 2021
%H Vincenzo Librandi, <a href="/A193657/b193657.txt">Table of n, a(n) for n = 0..200</a>
%H Dan Daly and Lara Pudwell, <a href="http://faculty.valpo.edu/lpudwell/slides/sandiego2013.pdf">Pattern avoidance in rook monoids</a>, Special Session on Patterns in Permutations and Words, Joint Mathematics Meetings, 2013. - From _N. J. A. Sloane_, Feb 03 2013
%F E.g.f.: (exp(x)-x)/(x-1)^2. - _Vaclav Kotesovec_, Nov 20 2012
%F a(n) ~ n!*n*(e-1). - _Vaclav Kotesovec_, Nov 20 2012
%F a(n) = 1-n*Gamma(n+1)+e*n*Gamma(n+1,1). - _Peter Luschny_, May 30 2014
%F a(n) +(-n-2)*a(n-1) +(n-1)*a(n-2)=0. - _R. J. Mathar_, May 30 2014
%F From _Peter Bala_, Feb 10 2020: (Start)
%F a(n) = n*A002627(n) + 1.
%F a(n) = A114870(n) + n!.
%F a(n) = A296964(n+1) - A296964(n) for n >= 2.
%F a(1) = 2 and a(n) = (n^2*a(n-1) - 1)/(n - 1) for n >= 2. See A082425 for solutions to this recurrence with different starting values.
%F Also, a(0) = 1 and a(n) = n*( a(n-1) + ... + a(0) ) + 1 for n >= 1.
%F Second column of A176305. (End)
%p a := n -> 1-n*GAMMA(n+1)+exp(1)*n*GAMMA(n+1,1):
%p seq(simplify(a(n)), n=0..9); # _Peter Luschny_, May 30 2014
%t q[n_, k_] := n + k + 1; (* A094727 *)
%t r[0] = 1; r[k_] := Sum[q[k - 1, i] r[k - 1 - i], {i, 0, k - 1}]
%t p[n_, k_] := 1
%t v[n_] := Sum[p[n, k] r[n - k], {k, 0, n}]
%t Table[v[n], {n, 0, 18}] (* A193657 *)
%t TableForm[Table[q[i, k], {i, 0, 4}, {k, 0, i}]]
%t Table[r[k], {k, 0, 8}] (* A193668 *)
%t TableForm[Table[p[n, k], {n, 0, 4}, {k, 0, 4}]]
%t CoefficientList[Series[(E^x-x)/(x-1)^2,{x,0,20}],x]*Range[0,20]! (* _Vaclav Kotesovec_, Nov 20 2012 *)
%o (PARI) a(n) = { sum(k=0, n, if (k <= n-2, binomial(n,k)*(k+1)!, binomial(n,k)^2*k!));} \\ _Michel Marcus_, Feb 07 2013
%o (Sage)
%o def A193657():
%o a = 2; b = 7; c = 31; n = 3
%o yield 1
%o while True:
%o yield a
%o n += 1
%o a,b,c = b,c,((n-2)^2*a+2*(1+n-n^2)*b+(3*n+n^2-2)*c)/n
%o a = A193657(); [next(a) for n in range(19)] # _Peter Luschny_, May 30 2014
%Y Cf. A193649, A094727, A082425, A114870, A176305, A296964.
%K nonn
%O 0,2
%A _Clark Kimberling_, Aug 02 2011
%E Simpler definition by _Peter Luschny_, May 30 2014
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