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A032123 Number of 2n-bead black-white reversible strings with n black beads. 2
1, 1, 4, 10, 38, 126, 472, 1716, 6470, 24310, 92504, 352716, 1352540, 5200300, 20060016, 77558760, 300546630, 1166803110, 4537591960, 17672631900, 68923356788, 269128937220, 1052049834576, 4116715363800 (list; graph; refs; listen; history; text; internal format)
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

n odd: C(2n-1, n-1); n even: C(2n-1, n-1) + C(n-1, n/2-1)

"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: 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.

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

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Last modified June 28 05:07 EDT 2017. Contains 288813 sequences.