A032123 Number of 2n-bead black-white reversible strings with n black beads.

1, 1, 4, 10, 38, 126, 472, 1716, 6470, 24310, 92504, 352716, 1352540, 5200300, 20060016, 77558760, 300546630, 1166803110, 4537591960, 17672631900, 68923356788, 269128937220, 1052049834576, 4116715363800, 16123803193628, 63205303218876, 247959271674352, 973469712824056

Offset

0,3

Comments

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]

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

C. G. Bower, Transforms (2)

N. J. A. Sloane, Classic Sequences

Formula

a(2n+1) = binomial(4n+1,2n) = A002458(n). a(2n) = binomial(4n-1,2n-1)+binomial(2n-1,n-1), n>0.

"BIK[ n ](2n-1)" (reversible, indistinct, unlabeled, n parts, 2n-1 elements) transform of 1, 1, 1, 1...

E.g.f.: exp(x)*cosh(x)*BesselI(0, 2*x). - Vladeta Jovovic, Apr 07 2005

G.f.: (1/2)*((1-4*x)^(-1/2)+(1-4*x^2)^(-1/2)). - Mark van Hoeij, Oct 30 2011

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

a(n) ~ 2^(2*n-1)/sqrt(Pi*n). - Vaclav Kotesovec, Mar 29 2014

Mathematica

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 *)

Crossrefs

Central column of Losanitsch's triangle A034851.

Cf. A002458 (bisection).

Sequence in context: A197051 A149191 A149192 * A149193 A149194 A149195

Adjacent sequences: A032120 A032121 A032122 * A032124 A032125 A032126

Keyword

nonn

Author

Christian G. Bower

Status

approved