A175124 - OEIS

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A175124

A symmetric triangle, with sum the large Schroeder numbers

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.` `From Peter Bala, Sep 30 2011: (Start)` `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. (End)`
	LINKS	`Table of n, a(n) for n=1..55.` `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.`
	FORMULA	`G.f. is the composition inverse of P(1-abP^2)/(1+aP)/(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+4bw+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-ab*_Z^2)); expand(taylor(f, x, 4));`
	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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Last modified May 22 09:43 EDT 2013. Contains 225519 sequences.