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

Author

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

Keywords

• easy
• nonn

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.

Crossrefs

Cf. A211419.

Programs

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 *)

Formula

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