A241907 - OEIS

A241907

a(n) = floor( Catalan(2*n) / Catalan(n)^2 ).

1, 2, 3, 5, 7, 9, 11, 14, 17, 20, 23, 26, 29, 33, 36, 40, 44, 48, 52, 56, 60, 65, 69, 74, 78, 83, 88, 93, 98, 103, 108, 114, 119, 124, 130, 136, 141, 147, 153, 159, 165, 171, 177, 183, 189, 196, 202, 209, 215, 222, 229, 235, 242, 249, 256, 263, 270, 277, 284, 292, 299, 306, 314, 321, 329, 336, 344, 352, 360, 367, 375, 383, 391, 399, 408, 416, 424, 432, 441, 449, 457, 466, 474, 483, 492, 500, 509, 518, 527, 536, 545, 554, 563, 572, 581, 590, 599, 609, 618, 627, 637, 646, 656, 665, 675, 685, 694, 704, 714, 724, 734, 744, 753, 764, 774, 784, 794, 804, 814, 825, 835, 845, 856, 866, 877, 887, 898, 908, 919, 930, 940, 951, 962, 973, 984, 995, 1006

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OFFSET

0,2

COMMENTS

This sequence is (roughly) the relative size of the Jones monoid J_n to its minimal ideal. Equivalently, this is roughly the reciprocal of the proportion of Dyck words of length 4n which can be factorized into two Dyck words, each of length 2n.

LINKS

Table of n, a(n) for n=0..136.

Wikipedia, Dyck word

FORMULA

a(n) = floor( Catalan( 2*n ) / Catalan(n)^2 ).

MAPLE

Digits:=200:

C:=n->binomial(2*n, n)/(n+1); f:=n->floor(C(2*n)/C(n)^2); [seq(f(n), n=0..100)]; # N. J. A. Sloane, May 21 2014

MATHEMATICA

Table[Floor[CatalanNumber[2n]/CatalanNumber[n]^2], {n, 0, 140}] (* Harvey P. Dale, Oct 04 2015 *)

CROSSREFS

Cf. A000108.