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A219614 Number of ways to put n labeled objects into n labeled boxes so that no two nonempty boxes are adjacent. 1
1, 1, 2, 9, 46, 335, 2786, 28357, 325382, 4280859, 62437882, 1010306825, 17852477006, 343275422503, 7120802805650, 158697470231757, 3778977532041430, 95794295907958547, 2574920565897373610, 73164585387874543057, 2191028450841437523230, 68974613397532849153311 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..200

V. Kotesovec, Interesting asymptotic formulas for binomial sums, Jun 09 2013

FORMULA

a(n) = Sum_{k>=0} binomial(n-k+1,k)*Stirling2(n,k)*k!.

Limit n->infinity (a(n)/n!)^(1/n) = (3*r^2-3*r+1)/(1-2*r) = 1.53445630931668421506236..., where r = 0.410751485627... is the root of the equation (1-2*r)^2 + r*(1-3*r+3*r^2)*LambertW(-exp(-1/r)/r) = 0. - Vaclav Kotesovec, Dec 08 2012

EXAMPLE

a(3) = 9 because we have: (1,1,3), (1,3,1), (1,3,3), (3,1,1), (3,1,3), (3,3,1), (1,1,1), (2,2,2), (3,3,3).

MAPLE

with (combinat):

a:= n-> add(stirling2(n, k)*k! *binomial(n-k+1, k), k=0..ceil(n/2)):

seq (a(n), n=0..30);  # Alois P. Heinz, Dec 06 2012

MATHEMATICA

Table[Sum[Binomial[n-k+1, k]StirlingS2[n, k]k!, {k, 0, n}], {n, 0, 20}]

(* Program for numerical value of the limit (a(n)/n!)^(1/n) *) (3*r^2-3*r+1)/(1-2*r)/.FindRoot[(1-2*r)^2+r*(1-3*r+3*r^2)*LambertW[-E^(-1/r)/r]==0, {r, 1/2}, WorkingPrecision->100] (* Vaclav Kotesovec, Dec 08 2012 *)

CROSSREFS

Cf. A120368.

Sequence in context: A218045 A161798 A134091 * A183166 A032331 A049371

Adjacent sequences:  A219611 A219612 A219613 * A219615 A219616 A219617

KEYWORD

nonn

AUTHOR

Geoffrey Critzer, Dec 06 2012

STATUS

approved

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Last modified May 16 01:55 EDT 2021. Contains 343937 sequences. (Running on oeis4.)