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%I #55 Jul 19 2023 02:11:15
%S 1,2,8,38,192,1002,5336,28814,157184,864146,4780008,26572086,
%T 148321344,830764794,4666890936,26283115038,148348809216,838944980514,
%U 4752575891144,26964373486406,153196621856192,871460014012682,4962895187697048,28292329581548718,161439727075246592
%N Row sums of A123160.
%C Coefficient of x^n in ((1 + x)/(1 - x))^n. - _Paul Barry_, Jan 20 2008
%C 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
%C Hankel transform is A180966. - _Paul Barry_, Sep 29 2010
%H Seiichi Manyama, <a href="/A123164/b123164.txt">Table of n, a(n) for n = 0..1000</a>
%H A. Laradji and A. Umar, <a href="http://dx.doi.org/10.1016/j.jalgebra.2003.10.023">A. Combinatorial results for semigroups of order-preserving partial transformations</a>, Journal of Algebra 278, (2004), 342-359.
%H A. Laradji and A. Umar, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL7/Umar/um.html">Combinatorial results for semigroups of order-decreasing partial transformations</a>, J. Integer Seq. 7 (2004), 04.3.8
%H A. Laradji and A. Umar, <a href="http://dx.doi.org/10.1080/00927870500442039">Asymptotic results for semigroups of order-preserving partial transformations</a> Comm. Algebra 34 (2006), 1071-1075. [From _Abdullahi Umar_, Oct 11 2008]
%H Huyile Liang, Yanni Pei, and Yi Wang, <a href="https://arxiv.org/abs/2302.11856">Analytic combinatorics of coordination numbers of cubic lattices</a>, arXiv:2302.11856 [math.CO], 2023. See p. 4.
%F a(n) = A122542(2*n,n). - _Philippe Deléham_, May 28 2007
%F a(n) = Sum_{k=0..n} C(n, k)*C(n+k-1, k). - _Paul Barry_, Aug 22 2007
%F (2*n-1)*(n+1)*a(n+1) = 4*(3*n^2-1)*a(n) - (2*n+1)*(n-1)*a(n-1) for n >= 1 with a(0) = 1 and a(1) = 2. - _Abdullahi Umar_, Aug 25 2008
%F a(n) = Jacobi_P(n, 0, -1, 3). - _Paul Barry_, Sep 27 2009
%F G.f.: (1 + x + sqrt(1 - 6*x + x^2))/(2*sqrt(1 - 6*x + x^2)). - _Paul Barry_, Sep 29 2010
%F From _Abdullahi Umar_, Oct 11 2008: (Start)
%F a(n+1) - a(n) = (2*n + 1)*A006318 (n >= 0);
%F 2*a(n) = (n + 1)*A006318(n) - (n - 1)*A006318(n-1) (n > 0). (End)
%F a(n) = Hypergeometric2F1([-n, n], [1], -1). - _Peter Luschny_, Aug 02 2014
%F a(n) ~ (1 + sqrt(2))^(2*n) / (2^(3/4) * sqrt(Pi*n)). - _Vaclav Kotesovec_, Feb 14 2021
%F From _Peter Bala_, Oct 07 2021: (Start)
%F a(n) = Sum_{k = 0..floor(n/2)} (-1)^k*C(n, k)*C(3*n-2*k-1, n-2*k).
%F a(p) == 2 (mod p^3) for prime p >= 5.
%F Conjecture: a(n*p^k) == a(n*p^(k-1)) mod( p^(3*k) ) for prime p >= 5 and all positive integers n and k. (End)
%t a[n_]:= a[n]= Sum[Binomial[n+k-1,k]*Binomial[n,k], {k,0,n}];
%t Table[a[n], {n,0,30}]
%o (Magma) [1] cat [n le 2 select 2*4^(n-1) else (4*(3*(n-1)^2-1)*Self(n-1) - (2*n-1)*(n-2)*Self(n-2))/((2*n-3)*(n)): n in [1..30]]; // _G. C. Greubel_, Jul 19 2023
%o (SageMath)
%o def A123164(n): return sum(binomial(n,j)*binomial(n+j-1,j) for j in range(n+1))
%o [A123164(n) for n in range(31)] # _G. C. Greubel_, Jul 19 2023
%Y Essentially identical to A002003.
%Y Cf. A006318, A122542, A180966.
%K nonn
%O 0,2
%A _Roger L. Bagula_, Oct 02 2006
%E Edited by _N. J. A. Sloane_, Oct 04 2006
%E Offset changed (a(0)=1) by _Michael Somos_, Feb 07 2011
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