OFFSET
0,2
COMMENTS
Let b(n) = S(d,n) be the coordination sequence of the lattice A_d. Then this sequence is a(n) = S(2n,n). See Conway-Sloane. The sequence is defined by Couveignes et al.
LINKS
R. H. Hardin and Colin Barker, Table of n, a(n) for n = 0..300 (terms up to n=26 from R. H. Hardin)
J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices VII: Coordination Sequences, Proc. Royal Soc. London, A453 (1997), 2369-2389 (pdf).
J.-M. Couveignes, T. Ezome, and R. Lercier, Elliptic periods and primality proving, arXiv:0810.2853 [math.NT], 2008-2009.
FORMULA
a(n) = S(2n,n) where S(d,n) = Sum_{k=0..d} C(d,k)^2*C(n-k+d-1,d-1) from formula (22) in Conway-Sloane.
a(n) ~ (1 + sqrt(2))^(4*n + 1/2) / (2^(5/4) * Pi * n). - Vaclav Kotesovec, Apr 10 2018
From Peter Bala, Dec 19 2020: (Start)
a(n) = Sum_{k = 0..n} C(2*n,n-k)^2 * C(2*n+k-1,k).
a(n) = Sum_{k = 1..n} C(2*n, k)*C(2*n+k, k)*C(n-1,k-1) for n >= 1.
a(n) = [x^n] P(2*n,(1 + x)/(1 - x)), where P(n,x) denotes the n-th Legendre polynomial. Cf. A103882.
a(n) = C(2*n,n)^2 * hypergeom([-n, -n, 2*n], [n+1, n+1], 1).
n^2*(2*n - 1)^2*(24*n^3 - 105*n^2 + 152*n - 73)*a(n) = (3264*n^7 - 20808*n^6 + 53900*n^5 - 73159*n^4 + 55963*n^3 - 24107*n^2 + 5436*n - 504)*a(n - 1) - (2*n - 1)*(2*n - 3)*(n - 2)^2*(24*n^3 - 33*n^2 + 14*n - 2)*a(n - 2).
Conjectural: for any prime p >= 5, a(n*p^k) == a(n*p^(k-1)) ( mod p^(3*k) ) for all positive integers n and k.
More generally, if r and s are positive integers, we conjecture that the same supercongruences hold for the sequence defined by [x^(r*n)] P(s*n,(1 + x)/(1 - x)). (End)
Even more generally, we conjecture that the same supercongruences hold for the sequence defined by [x^(r*n)] (1 + x)^(A*n) * (1 - x)^(B*n) * P(s*n,(1 + x)/(1 - x)), where A and B are integers. - Peter Bala, Mar 17 2023
a(n) = 2*Sum_{k = 0..2*n-1} (-1)^(k+1)*binomial(2*n-1, k)*binomial(n+k, k)* binomial(2*n+k-1, k) for n >= 1. - Peter Bala, Sep 25 2024
EXAMPLE
The a(1) = 6 sequences are (1,-1,0), (-1,1,0), (1,0,-1), (-1,0,1), (0,1,-1) and (0,-1,1).
MAPLE
S:=proc(d, n) add(binomial(d, k)^2*binomial(n-k+d-1, d-1), k=0..d); end proc; a:=n->S(2*n, n);
MATHEMATICA
Table[ Binomial[-1 + 3 n, -1 + 2 n] HypergeometricPFQ[{-2 n, -2 n, -n}, {1, 1 - 3 n}, 1], {n, 0, 10}] (* Eric W. Weisstein, Feb 10 2009 *)
PROG
(PARI) S(d, n) = sum(k=0, d, binomial(d, k)^2*binomial(n-k+d-1, d-1));
concat(1, vector(20, n, S(2*n, n))) \\ Colin Barker, Dec 24 2015
CROSSREFS
KEYWORD
easy,nice,nonn
AUTHOR
W. Edwin Clark, Feb 09 2009
EXTENSIONS
Formula incorrectly copied from A143699 removed by R. J. Mathar, Mar 11 2010
STATUS
approved