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
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.
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);
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)
MATHEMATICA
a[n_]:= a[n]= Sum[Binomial[n+k-1, k]*Binomial[n, k], {k, 0, n}];
Table[a[n], {n, 0, 30}]
PROG
(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
(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
CROSSREFS
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