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A256337 Number of R&E Family matchings on n edges. 1
1, 3, 14, 84, 592, 4624, 38527, 334520, 2985938, 27183525, 251204717, 2349231434, 22186989871, 211295835831, 2026765351330, 19563183525857, 189878734136185, 1852021863338181, 18143610295356357, 178450501118284066, 1761425423533593329, 17443012946883397306, 173247241040324621443, 1725404763442479917997, 17226646104539624029186, 172389875494663129310211, 1728819399958251503320772, 17371923980126335814068340, 174882805619639894274925366, 1763573883561574064590764255, 17813088298000097109198747848, 180193867968101208748329431620, 1825392351254024651444300421782, 18516189068195200519689971895953, 188058513067187124022632768762920, 1912273561123769149363329826421051, 19466808894886697821847017471202077, 198381665775949932177580813656191187, 2023702201274806730119560173885757279, 20663697938590662344370538856632433182 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The R&E Family of matchings is the largest family of matchings formed by repeated edge pinnings, edge inflations 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..40.

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

E. Rivas and S.R. Eddy, A dynamic programing algorithm for RNA structure prediction including pseudoknots, J. Mol. Biol., 285, (1999), 2053-2068.

FORMULA

G.f. f satisfies 2*x^(30)*f^(60) - 2*x^(29)*f^(59) - 38*x^(29)*f^(58) + 43*x^(28)*f^(57) + 342*x^(28)*f^(56) - 452*x^(27)*f^(55) - 1952*x^(27)*f^(54) + 3104*x^(26)*f^(53) + 8030*x^(26)*f^(52) - 15740*x^(25)*f^(51) - 25796*x^(25)*f^(50) + 63234*x^(24)*f^(49) + 68385*x^(24)*f^(48) - 210718*x^(23)*f^(47) - 154085*x^(23)*f^(46) + 600645*x^(22)*f^(45) + 295081*x^(22)*f^(44) - 1493939*x^(21)*f^(43) - (465768*x^(21)-30*x^(20))*f^(42) + 3281450*x^(20)*f^(41) + (556761*x^(20)-535*x^(19))*f^(40) - 6407159*x^(19)*f^(39) - (345063*x^(19)-4503*x^(18))*f^(38) + 11148795*x^(18)*f^(37) - (486467*x^(18)+23656*x^(17))*f^(36) - 17269156*x^(17)*f^(35) + (2247622*x^(17)+86500*x^(16))*f^(34) + 23701163*x^(16)*f^(33) - (5008690*x^(16)+232556*x^(15))*f^(32) - (28606954*x^(15)-2*x^(14))*f^(31) + (8286505*x^(15)+473426*x^(14))*f^(30) + (30112233*x^(14)-33*x^(13))*f^(29) - (10981985*x^(14)+739870*x^(13))*f^(28) - (27473085*x^(13)-241*x^(12))*f^(27) + (11935909*x^(13)+887762*x^(12))*f^(26) + (21720351*x^(12)-1046*x^(11))*f^(25) - (10798494*x^(12)+802751*x^(11))*f^(24) - (15014115*x^(11)-3036*x^(10))*f^(23) + (8290091*x^(11)+514052*x^(10))*f^(22) + (9236626*x^(10)-6259*x^(9))*f^(21) - (5538246*x^(10)+180970*x^(9))*f^(20) - (5149642*x^(9)-9471*x^(8))*f^(19) + (3290018*x^(9)-42243*x^(8))*f^(18) + (2610905*x^(8)-10692*x^(7))*f^(17) - (1741070*x^(8)-114094*x^(7))*f^(16) - (1180928*x^(7)-9042*x^(6))*f^(15) + (800928*x^(7)-91214*x^(6))*f^(14) + (460434*x^(6)-5687*x^(5))*f^(13) - (307889*x^(6)-46482*x^(5))*f^(12) - (148829*x^(5)-2607*x^(4))*f^(11) + (94979*x^(5)-16566*x^(4))*f^(10) + (38177*x^(4)-838*x^(3))*f^(9) - (22576*x^(4)-4140*x^(3))*f^(8) - (7344*x^(3)-176*x^(2))*f^(7) + (3919*x^(3)-695*x^(2))*f^(6) + (981*x^(2)-21*x)*f^(5) - (458*x^(2)-70*x)*f^(4) - (81*x-1)*f^(3) + (32*x-3)*f^(2) + 3*f - 1 = 0.

EXAMPLE

a(6)=4624 because, of the 4659 matchings on 6 edges which can be drawn in the plane, 65 do not lie in the R&E Family. In canonical sequence form, two of these missing matchings are given by 123143546526 and 123145643625.

MAPLE

f := RootOf(2*x^(30)*_Z^(60) - 2*x^(29)*_Z^(59) - 38*x^(29)*_Z^(58) + 43*x^(28)*_Z^(57) + 342*x^(28)*_Z^(56) - 452*x^(27)*_Z^(55) - 1952*x^(27)*_Z^(54) + 3104*x^(26)*_Z^(53) + 8030*x^(26)*_Z^(52) - 15740*x^(25)*_Z^(51) - 25796*x^(25)*_Z^(50) + 63234*x^(24)*_Z^(49) + 68385*x^(24)*_Z^(48) - 210718*x^(23)*_Z^(47) - 154085*x^(23)*_Z^(46) + 600645*x^(22)*_Z^(45) + 295081*x^(22)*_Z^(44) - 1493939*x^(21)*_Z^(43) - (465768*x^(21)-30*x^(20))*_Z^(42) + 3281450*x^(20)*_Z^(41) + (556761*x^(20)-535*x^(19))*_Z^(40) - 6407159*x^(19)*_Z^(39) - (345063*x^(19)-4503*x^(18))*_Z^(38) + 11148795*x^(18)*_Z^(37) - (486467*x^(18)+23656*x^(17))*_Z^(36) - 17269156*x^(17)*_Z^(35) + (2247622*x^(17)+86500*x^(16))*_Z^(34) + 23701163*x^(16)*_Z^(33) - (5008690*x^(16)+232556*x^(15))*_Z^(32) - (28606954*x^(15)-2*x^(14))*_Z^(31) + (8286505*x^(15)+473426*x^(14))*_Z^(30) + (30112233*x^(14)-33*x^(13))*_Z^(29) - (10981985*x^(14)+739870*x^(13))*_Z^(28) - (27473085*x^(13)-241*x^(12))*_Z^(27) + (11935909*x^(13)+887762*x^(12))*_Z^(26) + (21720351*x^(12)-1046*x^(11))*_Z^(25) - (10798494*x^(12)+802751*x^(11))*_Z^(24) - (15014115*x^(11)-3036*x^(10))*_Z^(23) + (8290091*x^(11)+514052*x^(10))*_Z^(22) + (9236626*x^(10)-6259*x^(9))*_Z^(21) - (5538246*x^(10)+180970*x^(9))*_Z^(20) - (5149642*x^(9)-9471*x^(8))*_Z^(19) + (3290018*x^(9)-42243*x^(8))*_Z^(18) + (2610905*x^(8)-10692*x^(7))*_Z^(17) - (1741070*x^(8)-114094*x^(7))*_Z^(16) - (1180928*x^(7)-9042*x^(6))*_Z^(15) + (800928*x^(7)-91214*x^(6))*_Z^(14) + (460434*x^(6)-5687*x^(5))*_Z^(13) - (307889*x^(6)-46482*x^(5))*_Z^(12) - (148829*x^(5)-2607*x^(4))*_Z^(11) + (94979*x^(5)-16566*x^(4))*_Z^(10) + (38177*x^(4)-838*x^(3))*_Z^(9) - (22576*x^(4)-4140*x^(3))*_Z^(8) - (7344*x^(3)-176*x^(2))*_Z^(7) + (3919*x^(3)-695*x^(2))*_Z^(6) + (981*x^(2)-21*x)*_Z^(5) - (458*x^(2)-70*x)*_Z^(4) - (81*x-1)*_Z^(3) + (32*x-3)*_Z^(2) + 3*_Z - 1, 1); series(f, x=0, 40);

CROSSREFS

Cf. A256330, A256338

Sequence in context: A121687 A154757 A074535 * A256330 A190761 A005700

Adjacent sequences:  A256334 A256335 A256336 * A256338 A256339 A256340

KEYWORD

nonn

AUTHOR

Aziza Jefferson, Mar 29 2015

STATUS

approved

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Last modified April 19 06:30 EDT 2019. Contains 322237 sequences. (Running on oeis4.)