1,2

Coefficient of x^n in ((1+x)/(1-x))^n; - Paul Barry (pbarry(AT)wit.ie), Jan 20 2008

a(n) is also the number of order-preserving partial transformations (of an n-element chain). Equivalently, it is the order of the semigroup (monoid) of order-preserving partial transformations (of an n-element chain), PO sub n. [From A. Umar (aumarh(AT)squ.edu.om), Aug 25 2008]

Laradji, A. and Umar, A. Combinatorial results for semigroups of order-preserving partial transformations. Journal of Algebra 278, (2004), 342-359 [From A. Umar (aumarh(AT)squ.edu.om), Aug 25 2008]

Laradji, A. and Umar, A. Asymptotic results for semigroups of order-preserving partial transformations. Comm. Algebra 34 (2006), 1071-1075. [From A. Umar (aumarh(AT)squ.edu.om), Oct 11 2008]

a(n+1)=A122542(2*n,n). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), May 28 2007

a(n)=sum{k=0..n, C(n,k)C(n+k-1,k)}; - Paul Barry (pbarry(AT)wit.ie), Aug 22 2007

(2n-1)(n+1)a(n+1)= 4(3n^2-1)a(n)-(2n+1)(n-1)a(n-1), a(0)= 1, a(1)= 2 [From A. Umar (aumarh(AT)squ.edu.om), Aug 25 2008]

a(n)=Jacobi_P(n,-1,-1,3). [From Paul Barry (pbarry(AT)wit.ie), Sep 27 2009]

t[n_, m_] = If [n == m == 0, 1, n!*(n + m - 1)!/((n - m)!*(n - 1)!(m!)^2)]; a = Table[Sum[t[n, m], {m, 0, n}], {n, 0, 20}]; Flatten[a]

Essentially identical to A002003.

Contribution from A. Umar (aumarh(AT)squ.edu.om), Oct 11 2008: (Start)

A123164(n+1) - A123164(n) = (2n+1)A006318 (n>=0);

2A123164(n) = (n+1)A006318 - (n-1)A006318 (n>0). (End)

Sequence in context: A053520 A112738 A155609 this_sequence A002003 A059423 A112109

Adjacent sequences: A123161 A123162 A123163 this_sequence A123165 A123166 A123167

nonn

Roger Bagula (rlbagulatftn(AT)yahoo.com), Oct 02 2006

Edited by N. J. A. Sloane (njas(AT)research.att.com), Oct 04 2006