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 A123164 Row sums of A123160. 6
 1, 2, 8, 38, 192, 1002, 5336, 28814, 157184, 864146, 4780008, 26572086, 148321344, 830764794, 4666890936, 26283115038, 148348809216, 838944980514, 4752575891144, 26964373486406, 153196621856192 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Coefficient of x^n in ((1+x)/(1-x))^n. - Paul Barry, 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. - Abdullahi Umar, Aug 25 2008 Hankel transform is A180966. - Paul Barry, Sep 29 2010 LINKS A. Laradji and A. Umar, A. Combinatorial results for semigroups of order-preserving partial transformations, Journal of Algebra 278, (2004), 342-359. A. Laradji, and A. Umar, Combinatorial results for semigroups of order-decreasing partial transformations, J. Integer Seq. 7 (2004), 04.3.8 A. Laradji and A. Umar, Asymptotic results for semigroups of order-preserving partial transformations Comm. Algebra 34 (2006), 1071-1075. [From Abdullahi Umar, Oct 11 2008] FORMULA a(n)=A122542(2*n,n). - Philippe Deléham, May 28 2007 a(n)=sum{k=0..n, C(n,k)C(n+k-1,k)}. - Paul Barry, 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. - Abdullahi Umar, Aug 25 2008 a(n)=Jacobi_P(n,0,-1,3). - Paul Barry, Sep 27 2009 G.f.: (1+x+sqrt(1-6x+x^2))/(2*sqrt(1-6x+x^2)). - Paul Barry, Sep 29 2010 From Abdullahi Umar, Oct 11 2008: (Start) a(n+1) - a(n) = (2n+1)A006318 (n>=0); 2*a(n) = (n+1)A006318(n) - (n-1)A006318(n-1) (n>0). (End) a(n) = Hyper2F1([-n, n], [1], -1). - Peter Luschny, Aug 02 2014 MATHEMATICA 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] CROSSREFS Essentially identical to A002003. Sequence in context: A155609 A220542 A199213 * A002003 A264229 A059423 Adjacent sequences:  A123161 A123162 A123163 * A123165 A123166 A123167 KEYWORD nonn AUTHOR Roger L. Bagula, Oct 02 2006 EXTENSIONS Edited by N. J. A. Sloane, Oct 04 2006 Offset changed (a(0)=1) by Michael Somos, Feb 07 2011 STATUS approved

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Last modified October 15 17:12 EDT 2018. Contains 316237 sequences. (Running on oeis4.)