A193657 First difference of A002627.

1, 2, 7, 31, 165, 1031, 7423, 60621, 554249, 5611771, 62353011, 754471433, 9876716941, 139097096919, 2097156230471, 33704296561141, 575219994643473, 10389911153247731, 198019483156015579, 3971390745517868001, 83608226221428800021, 1843561388182505040463

Offset

0,2

Comments

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.

Number of n X n rook placements avoiding the pattern 001. - N. J. A. Sloane, Feb 04 2013

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

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Dan Daly and Lara Pudwell, Pattern avoidance in rook monoids, Special Session on Patterns in Permutations and Words, Joint Mathematics Meetings, 2013. - From N. J. A. Sloane, Feb 03 2013

Formula

E.g.f.: (exp(x)-x)/(x-1)^2. - Vaclav Kotesovec, Nov 20 2012

a(n) ~ n!*n*(e-1). - Vaclav Kotesovec, Nov 20 2012

a(n) = 1-n*Gamma(n+1)+e*n*Gamma(n+1,1). - Peter Luschny, May 30 2014

a(n) +(-n-2)*a(n-1) +(n-1)*a(n-2)=0. - R. J. Mathar, May 30 2014

From Peter Bala, Feb 10 2020: (Start)

a(n) = n*A002627(n) + 1.

a(n) = A114870(n) + n!.

a(n) = A296964(n+1) - A296964(n) for n >= 2.

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.

Also, a(0) = 1 and a(n) = n*( a(n-1) + ... + a(0) ) + 1 for n >= 1.

Second column of A176305. (End)

Maple

a := n -> 1-n*GAMMA(n+1)+exp(1)*n*GAMMA(n+1, 1):
seq(simplify(a(n)), n=0..9); # Peter Luschny, May 30 2014

Mathematica

q[n_, k_] := n + k + 1;  (* A094727 *)
r[0] = 1; r[k_] := Sum[q[k - 1, i] r[k - 1 - i], {i, 0, k - 1}]
p[n_, k_] := 1
v[n_] := Sum[p[n, k] r[n - k], {k, 0, n}]
Table[v[n], {n, 0, 18}]    (* A193657 *)
TableForm[Table[q[i, k], {i, 0, 4}, {k, 0, i}]]
Table[r[k], {k, 0, 8}]  (* A193668 *)
TableForm[Table[p[n, k], {n, 0, 4}, {k, 0, 4}]]
CoefficientList[Series[(E^x-x)/(x-1)^2, {x, 0, 20}], x]*Range[0, 20]! (* Vaclav Kotesovec, Nov 20 2012 *)

Prog

(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
(Sage)
def A193657():
    a = 2; b = 7; c = 31; n = 3
    yield 1
    while True:
        yield a
        n += 1
        a, b, c = b, c, ((n-2)^2*a+2*(1+n-n^2)*b+(3*n+n^2-2)*c)/n
a = A193657(); [next(a) for n in range(19)] # Peter Luschny, May 30 2014

Crossrefs

Cf. A193649, A094727, A082425, A114870, A176305, A296964.

Sequence in context: A002872 A105216 A260532 * A007164 A321208 A005977

Adjacent sequences: A193654 A193655 A193656 * A193658 A193659 A193660

Keyword

nonn

Author

Clark Kimberling, Aug 02 2011

Extensions

Simpler definition by Peter Luschny, May 30 2014

Status

approved