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

Offset: 0

Author

Nick Loughlin, May 01 2014

Keywords

• nonn
• easy

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

• Wikipedia, Dyck word

Crossrefs

Cf. A000108.

Programs

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 *)

Formula

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

Extensions

Corrected by Harvey P. Dale, Oct 04 2015