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%I #37 Feb 15 2023 09:35:45
%S 1,1,4,10,38,126,472,1716,6470,24310,92504,352716,1352540,5200300,
%T 20060016,77558760,300546630,1166803110,4537591960,17672631900,
%U 68923356788,269128937220,1052049834576,4116715363800,16123803193628,63205303218876,247959271674352,973469712824056
%N Number of 2n-bead black-white reversible strings with n black beads.
%C It appears that a(n) is also the number of quivers in the mutation class of affine B_n or affine type C_n for n>=2. [_Christian Stump_, Nov 02 2010]
%H G. C. Greubel, <a href="/A032123/b032123.txt">Table of n, a(n) for n = 0..1000</a>
%H C. G. Bower, <a href="/transforms2.html">Transforms (2)</a>
%H N. J. A. Sloane, <a href="/classic.html#LOSS">Classic Sequences</a>
%F a(2n+1) = binomial(4n+1,2n) = A002458(n). a(2n) = binomial(4n-1,2n-1)+binomial(2n-1,n-1), n>0.
%F "BIK[ n ](2n-1)" (reversible, indistinct, unlabeled, n parts, 2n-1 elements) transform of 1, 1, 1, 1...
%F E.g.f.: exp(x)*cosh(x)*BesselI(0, 2*x). - _Vladeta Jovovic_, Apr 07 2005
%F G.f.: (1/2)*((1-4*x)^(-1/2)+(1-4*x^2)^(-1/2)). - _Mark van Hoeij_, Oct 30 2011
%F Conjecture: D-finite with recurrence n*(n-1)*a(n) -2*(n-1)*(3*n-4)*a(n-1) +4*(2*n^2-14*n+19)*a(n-2) +8*(n^2+5*n-19)*a(n-3) -16*(n-3)*(3*n-10)*a(n-4) +32*(n-4)*(2*n-9)*a(n-5)=0, n>5. - _R. J. Mathar_, Nov 09 2013
%F a(n) ~ 2^(2*n-1)/sqrt(Pi*n). - _Vaclav Kotesovec_, Mar 29 2014
%t With[{nn = 50}, CoefficientList[Series[Exp[x]*Cosh[x]*BesselI[0, 2*x], {x, 0, nn}], x] Range[0, nn]!] (* _G. C. Greubel_, Feb 15 2017 *)
%Y Central column of Losanitsch's triangle A034851.
%Y Cf. A002458 (bisection).
%K nonn
%O 0,3
%A _Christian G. Bower_