A123164 Row sums of A123160.

1, 2, 8, 38, 192, 1002, 5336, 28814, 157184, 864146, 4780008, 26572086, 148321344, 830764794, 4666890936, 26283115038, 148348809216, 838944980514, 4752575891144, 26964373486406, 153196621856192, 871460014012682, 4962895187697048, 28292329581548718, 161439727075246592

Offset: 0

Author

Roger L. Bagula, Oct 02 2006

Keywords

• nonn

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

• Seiichi Manyama, Table of n, a(n) for n = 0..1000
• 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]
• Huyile Liang, Yanni Pei, and Yi Wang, Analytic combinatorics of coordination numbers of cubic lattices, arXiv:2302.11856 [math.CO], 2023. See p. 4.

Crossrefs

Essentially identical to A002003.

Cf. A006318, A122542, A180966.

Programs

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

Mathematica

a[n_]:= a[n]= Sum[Binomial[n+k-1,k]*Binomial[n,k], {k,0,n}];
Table[a[n], {n,0,30}]

SageMath

def A123164(n): return sum(binomial(n,j)*binomial(n+j-1,j) for j in range(n+1))
[A123164(n) for n in range(31)] # G. C. Greubel, Jul 19 2023

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

(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

a(n) = Jacobi_P(n, 0, -1, 3). - Paul Barry, Sep 27 2009

G.f.: (1 + x + sqrt(1 - 6*x + x^2))/(2*sqrt(1 - 6*x + x^2)). - Paul Barry, Sep 29 2010

From Abdullahi Umar, Oct 11 2008: (Start)

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

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

a(n) = Hypergeometric2F1([-n, n], [1], -1). - Peter Luschny, Aug 02 2014

a(n) ~ (1 + sqrt(2))^(2*n) / (2^(3/4) * sqrt(Pi*n)). - Vaclav Kotesovec, Feb 14 2021

From Peter Bala, Oct 07 2021: (Start)

a(n) = Sum_{k = 0..floor(n/2)} (-1)^k*C(n, k)*C(3*n-2*k-1, n-2*k).

a(p) == 2 (mod p^3) for prime p >= 5.

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)

Extensions

Edited by N. J. A. Sloane, Oct 04 2006

Offset changed (a(0)=1) by Michael Somos, Feb 07 2011