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A221957
Number of n X n rook placements avoiding the pattern 012.
1
1, 2, 7, 31, 159, 921, 5988, 43632, 355491, 3223729, 32329668, 355979064, 4273100846, 55555511298, 777797216472, 11667035805840, 186672873433635, 3173440015174905, 57121924810715940, 1085316589076234760, 21706331850447959610, 455832969128536089030
OFFSET
0,2
LINKS
Dan Daly and Lara Pudwell, Pattern avoidance in rook monoids, Special Session on Patterns in Permutations and Words, Joint Mathematics Meetings, 2013.
FORMULA
From Vaclav Kotesovec, Feb 07 2013: (Start)
E.g.f.: 1/2*(exp(2*x)*BesselI(0,2*x)+1)/(1-x).
Recurrence: n*a(n) = (n^2+4*n-2)*a(n-1)-2*(n-1)*(2*n-1)*a(n-2).
a(n) ~ c * n!, where c = (exp(2)*BesselI(0,2)+1)/2 = 8.921991840629494...
(End)
c = 1 + Sum_{k>=0} binomial(2*k+1,k) / (k+1)!. - Vaclav Kotesovec, Jul 19 2021
MATHEMATICA
CoefficientList[Series[1/2*(Exp[2*x]*BesselI[0, 2*x]+1)/(1-x), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Feb 07 2013 *)
PROG
(PARI) a(n) = {sum(k=0, n, if (k==n, n!, sum(j=1, k+1, binomial(n-j, n-k-1)*binomial(n, k)*binomial(k, j-1)*(j-1)!))); } \\ Michel Marcus, Feb 07 2013
CROSSREFS
Sequence in context: A030882 A273957 A221958 * A030966 A009132 A125275
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Feb 04 2013
EXTENSIONS
More terms from Michel Marcus, Feb 07 2013
STATUS
approved