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A256338 Number of A&U Family matchings on n edges. 1
1, 3, 14, 81, 526, 3655, 26522, 198322, 1516296, 11794717, 93028387, 742192059, 5978650560, 48558821234, 397218622275, 3269629207524, 27061726430000, 225078993453143, 1880240716499975, 15768890757767329, 132719696885282352, 1120664726059889642, 9490737694928103944, 80593740187789336604, 686097231181385302494, 5854230604182513256777, 50058728487687099021228, 428893610758038945556024, 3681458291424994103104272, 31654643493605098603330050, 272617697673293256259943417, 2351397730980411031399548438, 20310185543805378949877753778, 175663385844074502933143530174, 1521230708939544454165789841800, 13189400713003422051741601456307, 114483609078595784724427186310842, 994773380472692869438699360298740, 8652545469871591210786412806190538 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The A&U Family of matchings is the largest family of matchings formed by pinning edges to the right, edge inflation by ladders and vertex insertions.

REFERENCES

A. Jefferson, The Substitution Decomposition of Matchings and RNA Secondary Structures, Ph. D. Dissertation, Univ. of Florida, Math., 2015.

LINKS

Table of n, a(n) for n=1..39.

T. Akutsu, Dynamic programming algorithms for RNA secondary structure prediction with pseudoknots, Discrete Appl. Math. 104(1-2), (2000), 45-62.

A. Condon, B. Davy, B. Rastegari, S. Zhao and F. Tarrant, RNA pseudoknotted structures, Theoret. Comput. Sci. 320(1), (2004), 35-50.

C. Saule, M. R├ęgnier, J.-M. Steyaert, and A. Denise, Counting RNA pseudoknotted structures, J. Comput. Biol. 18(10), (2011), 1339-1351.

Y. Uemura, A. Hasegawa, S. Kobayashi, and T. Yokomori,  Tree adjoining grammars for RNA structure prediction, Theoret. Comput. Sci. 210(2), (1999), 277-303.

FORMULA

G.f. f satisfies 2x^7f^14 - 2x^6f^13 - 3x^6f^12 + 7x^5f^11 + 3x^5f^10 - 16x^4f^9 + 2x^4f^8 + 18x^3f^7 - 7x^2f^5 + (-12x^3 + 2x^2)f^6 + (4x^2 - 5x)f^4 + 10xf^3 + (-5x+1)f^2 - 2f + 1 = 0.

EXAMPLE

a(4)=81 because of the 105 matchings on 4 edges which can be drawn in the plane, 24 do not lie in the A&U Family. Of these 24, only three lie in the R&E family. In canonical sequence form the three missing matchings are given by 12134324, 12324314, and 12343142.

MAPLE

f := RootOf(2*x^7*_Z^(14)-2*x^6*_Z^(13)-3*x^6*_Z^(12)+7*x^5*_Z^(11)+3*x^5*_Z^(10)-16*x^4*_Z^9+2*x^4*_Z^8+18*x^3*_Z^7-7*x^2*_Z^5+(-12*x^3+2*x^2)*_Z^6+(4*x^2-5*x)*_Z^4+10*x*_Z^3+(-5*x+1)*_Z^2-2*_Z+1); convert(series(f, x=0, 40), radical);

CROSSREFS

Cf. A256330, A256337.

Sequence in context: A009053 A202474 A256336 * A256331 A292875 A077054

Adjacent sequences:  A256335 A256336 A256337 * A256339 A256340 A256341

KEYWORD

nonn

AUTHOR

Aziza Jefferson, Mar 29 2015

STATUS

approved

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Last modified March 20 23:22 EDT 2019. Contains 321354 sequences. (Running on oeis4.)