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

0,2

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.

Sequence in context: A062427 A127721 A292620 * A065130 A023535 A056834

Adjacent sequences: A241904 A241905 A241906 * A241908 A241909 A241910

Keyword

nonn,easy

Author

Nick Loughlin, May 01 2014

Extensions

Corrected by Harvey P. Dale, Oct 04 2015

Status

approved