This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A193657 #49 Feb 04 2024 16:36:11
%S A193657 1,2,7,31,165,1031,7423,60621,554249,5611771,62353011,754471433,
%T A193657 9876716941,139097096919,2097156230471,33704296561141,575219994643473,
%U A193657 10389911153247731,198019483156015579,3971390745517868001,83608226221428800021,1843561388182505040463
%N A193657 First difference of A002627.
%C A193657 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 A193657 Number of n X n rook placements avoiding the pattern 001. - _N. J. A. Sloane_, Feb 04 2013
%C A193657 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 A193657 Vincenzo Librandi, <a href="/A193657/b193657.txt">Table of n, a(n) for n = 0..200</a>
%H A193657 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 A193657 E.g.f.: (exp(x)-x)/(x-1)^2. - _Vaclav Kotesovec_, Nov 20 2012
%F A193657 a(n) ~ n!*n*(e-1). - _Vaclav Kotesovec_, Nov 20 2012
%F A193657 a(n) = 1-n*Gamma(n+1)+e*n*Gamma(n+1,1). - _Peter Luschny_, May 30 2014
%F A193657 a(n) +(-n-2)*a(n-1) +(n-1)*a(n-2)=0. - _R. J. Mathar_, May 30 2014
%F A193657 From _Peter Bala_, Feb 10 2020: (Start)
%F A193657 a(n) = n*A002627(n) + 1.
%F A193657 a(n) = A114870(n) + n!.
%F A193657 a(n) = A296964(n+1) - A296964(n) for n >= 2.
%F A193657 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 A193657 Also, a(0) = 1 and a(n) = n*( a(n-1) + ... + a(0) ) + 1 for n >= 1.
%F A193657 Second column of A176305. (End)
%p A193657 a := n -> 1-n*GAMMA(n+1)+exp(1)*n*GAMMA(n+1,1):
%p A193657 seq(simplify(a(n)), n=0..9); # _Peter Luschny_, May 30 2014
%t A193657 q[n_, k_] := n + k + 1; (* A094727 *)
%t A193657 r[0] = 1; r[k_] := Sum[q[k - 1, i] r[k - 1 - i], {i, 0, k - 1}]
%t A193657 p[n_, k_] := 1
%t A193657 v[n_] := Sum[p[n, k] r[n - k], {k, 0, n}]
%t A193657 Table[v[n], {n, 0, 18}] (* A193657 *)
%t A193657 TableForm[Table[q[i, k], {i, 0, 4}, {k, 0, i}]]
%t A193657 Table[r[k], {k, 0, 8}] (* A193668 *)
%t A193657 TableForm[Table[p[n, k], {n, 0, 4}, {k, 0, 4}]]
%t A193657 CoefficientList[Series[(E^x-x)/(x-1)^2,{x,0,20}],x]*Range[0,20]! (* _Vaclav Kotesovec_, Nov 20 2012 *)
%o A193657 (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 A193657 (Sage)
%o A193657 def A193657():
%o A193657 a = 2; b = 7; c = 31; n = 3
%o A193657 yield 1
%o A193657 while True:
%o A193657 yield a
%o A193657 n += 1
%o A193657 a,b,c = b,c,((n-2)^2*a+2*(1+n-n^2)*b+(3*n+n^2-2)*c)/n
%o A193657 a = A193657(); [next(a) for n in range(19)] # _Peter Luschny_, May 30 2014
%Y A193657 Cf. A193649, A094727, A082425, A114870, A176305, A296964.
%K A193657 nonn
%O A193657 0,2
%A A193657 _Clark Kimberling_, Aug 02 2011
%E A193657 Simpler definition by _Peter Luschny_, May 30 2014