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A241907
a(n) = floor( Catalan(2*n) / Catalan(n)^2 ).
0
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.
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
KEYWORD
nonn,easy
AUTHOR
Nick Loughlin, May 01 2014
EXTENSIONS
Corrected by Harvey P. Dale, Oct 04 2015
STATUS
approved