A173781 a(n) is the smallest entry of the n-th column of the matrix of Super Catalan numbers S(m,n).

1, 2, 4, 10, 28, 72, 198, 572, 1560, 4420, 12920, 36176, 104006, 305900, 869400, 2521260, 7443720, 21360240, 62300700, 184410072, 532740208, 1560167752, 4626704368, 13432367520, 39457579590, 117177054540, 341487416088, 1005490725148, 2989296750440, 8737944347440, 25776935824948

Offset

0,2

Comments

Super Catalan number S(m,n) is [(2m)! (2n)! ] / [(m!) (n!) (m+n)! ], where m,n are nonnegative integers.

S(m,n) is a positive integer, but a combinatorial interpretation of S(m,n) is an open problem.

For each n, the sequence S(m,n) is decreasing then increasing, with minimum value at m = ceiling(n/3).

Our sequence is that list of values S( ceiling(n/3), n).

S(n,k-n) = C(2k,k) * C(k,n) / C(2k,2n). - Charlie Neder, Dec 27 2018

Links

Seiichi Manyama, Table of n, a(n) for n = 0..2099

Ira M. Gessel, Super ballot numbers, J. Symbolic Comp., 14 (1992), 179-194.

Ira M. Gessel and Guoce Xin, A Combinatorial Interpretation of the Numbers 6(2n)!/n!(n+2)!, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.3, 13 pp.

Formula

a(3*n) = A211419(n). - Peter Bala, Sep 24 2023

Mathematica

nn = 30; {1}~Join~Table[Min@ Map[Function[n, ((2 m)! (2 n)!)/((m!) (n!) (m + n)!)], Range@ nn], {m, nn}] (* Michael De Vlieger, Jul 16 2016 *)

Crossrefs

Cf. A211419.

Sequence in context: A321946 A244485 A128933 * A340825 A106362 A271896

Adjacent sequences: A173778 A173779 A173780 * A173782 A173783 A173784

Keyword

easy,nonn

Author

Joseph Alfano (jalfano(AT)assumption.edu), Feb 24 2010

Status

approved