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A193657 First difference of A002627. 3
1, 2, 7, 31, 165, 1031, 7423, 60621, 554249, 5611771, 62353011, 754471433, 9876716941, 139097096919, 2097156230471, 33704296561141, 575219994643473, 10389911153247731, 198019483156015579 (list; graph; refs; listen; history; text; internal format)
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

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

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(); [a.next() for n in range(19)] # Peter Luschny, May 30 2014

CROSSREFS

Cf. A193649, A094727.

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

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Last modified January 26 23:05 EST 2020. Contains 331289 sequences. (Running on oeis4.)