cp's OEIS Frontend

A078698 Number of ways to lace a shoe that has n pairs of eyelets such that each eyelet has at least one direct connection to the opposite side.

%I A078698 #33 Mar 11 2025 12:31:56
%S A078698 1,2,20,396,14976,907200,79315200,9551001600,1513528934400,
%T A078698 305106949324800,76296489615360000,23175289163980800000,
%U A078698 8404709419090575360000,3587225703492542791680000,1779970753996760560435200000,1016036270188884847558656000000,661106386935312429191528448000000
%N A078698 Number of ways to lace a shoe that has n pairs of eyelets such that each eyelet has at least one direct connection to the opposite side.
%C A078698 The lace is "directed": reversing the order of eyelets along the path counts as a different solution. It must begin and end at the extreme pair of eyelets,
%D A078698 C. A. Pickover, The Math Book, Sterling, NY, 2009; see p. 494.
%H A078698 A. Khrabrov and K. Kokhas, <a href="http://arxiv.org/abs/1505.06309">Points on a line, shoelace and dominoes</a>, arXiv:1505.06309 [math.CO], (23-May-2015).
%H A078698 Hugo Pfoertner, <a href="/A078698/a078698.f.txt">Fortran program</a>.
%H A078698 Hugo Pfoertner, <a href="/A078698/a078698.txt">Results for n=1..4</a>.
%H A078698 Hugo Pfoertner, <a href="/A078698/a078698_1.txt">Results for n=5</a>.
%H A078698 <a href="/index/La#lacings">Index entries for sequences related to shoe lacings</a>
%F A078698 Conjecture: a(n) = (n-1)!^2*A051286(n). - _Vladeta Jovovic_, Sep 14 2005 (correct, see the Khrabrov/Kokhas reference, _Joerg Arndt_, May 26 2015)
%e A078698 a(3) = 20: label the eyelets 1,2,3 from front to back on the left side then 4,5,6 from back to front on the right side. The lacings are: 124356 154326 153426 142536 145236 135246 and the following and their mirror images: 125346 124536 125436 152346 153246 152436 154236.
%e A078698 Examples for n=2,3,4 can be found following the FORTRAN program at given link.
%t A078698 a[n_] := (n-1)!^2 Sum[Binomial[n-k, k]^2, {k, 0, n/2}];
%t A078698 Array[a, 17] (* _Jean-François Alcover_, Jul 20 2018 *)
%o A078698 (Fortran) c Program provided at Pfoertner link
%Y A078698 Cf. A078700, A078702, A078602.
%K A078698 nonn
%O A078698 1,2
%A A078698 _Hugo Pfoertner_, Dec 18 2002
%E A078698 Terms a(9) and beyond (using A051286) from _Joerg Arndt_, May 26 2015