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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.

%I A231580 #41 Mar 24 2021 22:58:56
%S A231580 1,2,7,34,638,4876,220217,6885458,569311642,7515775348,197394815194,
%T A231580 78863079581996,886395722771204,848070074996694008,
%U A231580 222148423805582000341,33494470531439170224754,35665304857619152523926,280147437461017444466304484
%N 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.
%C A231580 For n=1 the length of the line to cover is equal to 2. There is only one way to cover it with 2-length 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 2-length segments. The first segment could be placed at 3 position with probability 1/3 in the following ways (xxoo, oxxo, ooxx). The second 2-length 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 4-length line when it is randomly sequentially filled by 2-length segments is equal to 2/3.
%H A231580 Vincenzo Librandi, <a href="/A231580/b231580.txt">Table of n, a(n) for n = 1..260</a>
%H A231580 Philipp O. Tsvetkov, <a href="https://doi.org/10.1038/s41598-020-77896-0">Stoichiometry of irreversible ligand binding to a one-dimensional lattice</a>, Scientific Reports, Springer Nature (2020) Vol. 10, Article number: 21308.
%F A231580 Numerator of f(n), where f(0)=1 and f(n) = Sum_{k=0..n-1} f(k)*f(n-k-1)/(2*n-1). - _Michael Somos_, Mar 01 2014
%e A231580 1, 2/3, 7/15, 34/105, 638/2835, 4876/31185, 220217/2027025, 6885458/91216125, 569311642/10854718875, 7515775348/206239658625, 197394815194/7795859096025, ...
%p A231580 A231580f := proc(n)
%p A231580     option remember;
%p A231580     if n <= 0 then
%p A231580         1;
%p A231580     else
%p A231580         add(procname(k)*procname(n-k-1),k=0..n-1)/(2*n-1) ;
%p A231580     end if;
%p A231580 end proc:
%p A231580 A231580 := proc(n)
%p A231580     numer(A231580f(n)) ;
%p A231580 end proc:
%p A231580 seq(A231580(n),n=1..30) ; # _R. J. Mathar_, Aug 28 2014
%t A231580 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);
%t A231580 f[{_}] = f[{_}, _] = 1;
%t A231580 f[ConstantArray[0, #], 2] & /@ Range[2, 20] // Numerator
%o A231580 (PARI) f=[1]; for(n=2, 25, f=concat(f, sum(k=1, n-1, (f[k]*f[n-k])) / (2*n-3))); f
%o A231580 vector(#f, k, numerator(f[k])) \\ _Colin Barker_, Jul 24 2014, for sequence shifted by 1 index
%Y A231580 Cf. A231634 (denominators).
%K A231580 nonn,frac
%O A231580 1,2
%A A231580 _Philipp O. Tsvetkov_, Nov 11 2013
%E A231580 Name edited by _Jon E. Schoenfield_, Nov 13 2018