

A231580


a(n) is the numerator of the probability that n segments of length 2, each placed randomly on a line segment of length 2n, will completely cover the line segment.


4



1, 2, 7, 34, 638, 4876, 220217, 6885458, 569311642, 7515775348, 197394815194, 78863079581996, 886395722771204, 848070074996694008, 222148423805582000341, 33494470531439170224754, 35665304857619152523926, 280147437461017444466304484
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OFFSET

1,2


COMMENTS

For n=1 the length of the line to cover is equal to 2. There is only one way to cover it with 2length segment and it will be the full cover. So, the probability is equal to 1. For n=2 the length of the line to cover is equal to 4. Let's start randomly and sequentially to cover it with 2length segments. The first segment could be placed at 3 position with probability 1/3 in the following ways (xxoo, oxxo, ooxx). The second 2length segment could be added only in the first and the last cases. So we have the following covers (xxxx, oxxo, xxxx). Thus the probability to find the full cover of 4length line when it is randomly sequentially filled by 2length segments is equal to 2/3.


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..260


FORMULA

Numerator of f(n), where f(0)=1 and f(n) = Sum_{k=0..n1} f(k)*f(nk1)/(2*n1).  Michael Somos, Mar 01 2014


EXAMPLE

1, 2/3, 7/15, 34/105, 638/2835, 4876/31185, 220217/2027025, 6885458/91216125, 569311642/10854718875, 7515775348/206239658625, 197394815194/7795859096025, ...


MAPLE

A231580f := proc(n)
option remember;
if n <= 0 then
1;
else
add(procname(k)*procname(nk1), k=0..n1)/(2*n1) ;
end if;
end proc:
A231580 := proc(n)
numer(A231580f(n)) ;
end proc:
seq(A231580(n), n=1..30) ; # R. J. Mathar, Aug 28 2014


MATHEMATICA

f[g_List, l_] := f[g, l] = Sum[f[g[[;; n]], l] f[g[[n + 1 ;; ]], l], {n, Length[g]  1}]/(Total[l + g]  2 l + 1);
f[{_}] = f[{_}, _] = 1;
f[ConstantArray[0, #], 2] & /@ Range[2, 20] // Numerator


PROG

(PARI) f=[1]; for(n=2, 25, f=concat(f, sum(k=1, n1, (f[k]*f[nk])) / (2*n3))); f
vector(#f, k, numerator(f[k])) \\ Colin Barker, Jul 24 2014, for sequence shifted by 1 index


CROSSREFS

Cf. A231634 (denominators).
Sequence in context: A222868 A222777 A222891 * A182408 A057298 A058941
Adjacent sequences: A231577 A231578 A231579 * A231581 A231582 A231583


KEYWORD

nonn,frac


AUTHOR

Philipp O. Tsvetkov, Nov 11 2013


EXTENSIONS

Name edited by Jon E. Schoenfield, Nov 13 2018


STATUS

approved



