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A156554 The number of integer sequences of length d = 2n+1 such that the sum of the terms is 0 and the sum of the absolute values of the terms is d-1. 24
1, 6, 110, 2562, 66222, 1815506, 51697802, 1511679210, 45076309166, 1364497268946, 41800229045610, 1292986222651646, 40317756506959050, 1265712901796074842, 39965073938276694002, 1268208750951634765562, 40419340092267053380782, 1293151592990764737265490 (list; graph; refs; listen; history; text; internal format)
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, (2008) (pdf.

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.

EXAMPLE

For n = 1 the a(n) = 6 sequences are (1,-1,0),(-1,1,0),(1,0,-1),(-1,0,1),(0,1,-1),(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

a(n) = A103881(2n, n).

Sequence in context: A119814 A227443 A050884 * A197325 A197978 A241702

Adjacent sequences:  A156551 A156552 A156553 * A156555 A156556 A156557

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

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Last modified March 25 01:30 EDT 2017. Contains 284036 sequences.