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 A175124 A symmetric triangle, with sum the large Schröder numbers. 2
 1, 1, 1, 1, 4, 1, 1, 10, 10, 1, 1, 20, 48, 20, 1, 1, 35, 161, 161, 35, 1, 1, 56, 434, 824, 434, 56, 1, 1, 84, 1008, 3186, 3186, 1008, 84, 1, 1, 120, 2100, 10152, 16840, 10152, 2100, 120, 1, 1, 165, 4026, 28050, 70807, 70807, 28050, 4026, 165, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS a(n) is the number of noncrossing plants in the n+1 polygon, with no right corner, according to the number of left and top corners. T(n,k) counts ordered complete binary trees with n leaves having k internal vertices colored black, the remaining n-1-k internal vertices colored white, and such that each vertex and its rightmost child have different colors. An example is given below. See Example 1.6.7 in [Drake] but note this triangle is not equal to A089447 as stated there. Compare with A196201. - Peter Bala, Sep 30 2011 LINKS B. Drake, An inversion theorem for labeled trees and some limits of areas under lattice paths, A dissertation presented to the Faculty of the Graduate School of Arts and Sciences of Brandeis University. Shishuo Fu, Z. Lin, J. Zeng, Two new unimodal descent polynomials, arXiv preprint arXiv:1507.05184 [math.CO], 2015. FORMULA G.f. is the composition inverse of P*(1-a*b*P^2)/(1+a*P)/(1+b*P). EXAMPLE 1; 1,1; 1,4,1; 1,10,10,1; Triangle begins n\k.|..1....2....3....4....5....6....7 = = = = = = = = = = = = = = = = = = = = ..1.|..1 ..2.|..1....1 ..3.|..1....4....1 ..4.|..1...10...10....1 ..5.|..1...20...48...20....1 ..6.|..1...35..161..161...35....1 ..7.|..1...56..434..824..434...56....1 ... Row 3: b^2+4*b*w+w^2. Internal vertices colored either b(lack) or w(hite); 3 uncolored leaf nodes shown as o. ..Weight.....b^2...........w^2. ..............b.............w....... ............./\............/\....... ............/..\........../..\...... ...........b....o........w....o..... ........../\............/\.......... ........./..\........../..\......... ........o....o........o....o........ .................................... ..Weight.......b*w. ........b...................w....... ......./\................../\....... ....../..\................/..\...... .....w....o..............b....o..... ..../\................../\.......... .../..\................/..\......... ..o....o...............o..o......... .................................... ........b..........w................ ......./\........./\................ ....../..\......./..\............... .....o....w.....o....b.............. ........../\........./\............. ........./..\......./..\............ ........o....o.....o....o........... .................................... MAPLE f:=RootOf((1+a*_Z)*(1+b*_Z)*x-_Z*(1-a*b*_Z^2)); expand(taylor(f, x, 4)); MATHEMATICA ab = InverseSeries[P*(1-a*b*P^2)/(1+a*P)/(1+b*P)+O[P]^12, P] // Normal // CoefficientList[#, P]&; (List @@@ ab) /. a|b -> 1 // Rest // Flatten (* Jean-François Alcover, Feb 23 2017 *) CROSSREFS Cf. A006318 (row sums), A196201. Sequence in context: A214398 A220860 A174043 * A089447 A082680 A056939 Adjacent sequences:  A175121 A175122 A175123 * A175125 A175126 A175127 KEYWORD nonn,tabl AUTHOR F. Chapoton, Feb 15 2010 STATUS approved

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