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

Author

Christian G. Bower

Keywords

• nonn

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

Crossrefs

Central column of Losanitsch's triangle A034851.

Cf. A002458 (bisection).

Programs

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 *)

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