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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

Table of n, a(n) for n=0..20.

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 A059423 A112109

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 August 28 13:24 EDT 2015. Contains 261122 sequences.