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A176006 The number of branching configurations of RNA (see Sankoff, 1985) with n or fewer hairpins. 1
1, 2, 4, 10, 32, 122, 516, 2322, 10880, 52466, 258564, 1296282, 6589728, 33887466, 175966212, 921353250, 4858956288, 25786112994, 137604139012, 737922992938, 3974647310112, 21493266631002, 116642921832964, 635074797251890 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]

Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.

David Sankoff, Simultaneous solution of the RNA folding, alignment and protosequence problems, SIAM J. Appl. Math 45(5) (1985), 810-825.

David Sankoff, Simultaneous solution of the RNA folding, alignment and protosequence problems, SIAM J. Appl. Math 45(5) (1985), 810-825.

FORMULA

G.f.: (3 - x - sqrt(1 - 6*x + x^2))/(2*(1 - x)).

Conjecture : n*a(n) +(9-7*n)*a(n-1) +(7*n-12)*a(n-2) +(3-n)*a(n-3)=0. - R. J. Mathar, Jul 24 2012

a(n) ~ 2^(1/4)*(3 + 2*sqrt(2))^n/(4*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 20 2012

EXAMPLE

For n = 3, the a(3) = 10 branching configurations with 3 or fewer hairpins are: unfolded, (), ()(), (()()), ()()(), (()())(), ()(()()), (()()()), ((()())()), and (()(()())).

MATHEMATICA

CoefficientList[Series[(3-x-Sqrt[1-6*x+x^2])/(2*(1-x)), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *)

PROG

(PARI) my(x='x+O('x^50)); Vec((3-x-sqrt(1-6*x+x^2))/(2*(1-x))) \\ G. C. Greubel, Mar 22 2017

CROSSREFS

The cumulative sums of A155069.

Sequence in context: A120017 A000736 A263663 * A263664 A263665 A001250

Adjacent sequences:  A176003 A176004 A176005 * A176007 A176008 A176009

KEYWORD

easy,nonn

AUTHOR

Lee A. Newberg, Apr 05 2010

STATUS

approved

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Last modified December 5 12:24 EST 2021. Contains 349557 sequences. (Running on oeis4.)