A109966 a(n) = 8^((n^2-n)/2).

1, 1, 8, 512, 262144, 1073741824, 35184372088832, 9223372036854775808, 19342813113834066795298816, 324518553658426726783156020576256, 43556142965880123323311949751266331066368, 46768052394588893382517914646921056628989841375232, 401734511064747568885490523085290650630550748445698208825344

Offset: 0

Author

Philippe Deléham, Sep 01 2005

Keywords

• nonn
• easy

Comments

Sequence given by the Hankel transform (see A001906 for definition) of A082147 = {1, 1, 9, 89, 945, 10577, 123129, 1476841, ...}; example: det([1, 1, 9, 89; 1, 9, 89, 945; 9, 89, 945, 10577; 89, 945, 10577, 123129]) = 8^6 = 262144.

The number of labeled multigraphs on n vertices such that (i) no self loops are allowed; (ii) all edges are painted in one of 3 colors; (iii) edges between any pair of vertices are painted in distinct colors. Note, this implies that there are at most 3 edges between any vertex pair. Also note there is no restriction on the color of edges incident to a common vertex. - Geoffrey Critzer, Jan 14 2020

Crossrefs

Cf. A006125, A047656, A053763, A053764, A059435, A082147, A109345, A109354, A109493.

Programs

Magma

[2^(3*Binomial(n,2)): n in [0..10]]; // G. C. Greubel, Feb 05 2018

Mathematica

Table[2^(3*Binomial[n,2]),{n,0,10}] (* Geoffrey Critzer, Nov 10 2011 *)

PARI

a(n)=8^binomial(n,2) \\ Charles R Greathouse IV, Jan 17 2012

Formula

a(n+1) is the determinant of n X n matrix M_(i, j) = binomial(8i, j).

Hankel transform of A059435. - Philippe Deléham, Sep 03 2006

Extensions

a(10) corrected and a(11), a(12) from Georg Fischer, Apr 01 2022