%I #19 Jan 09 2024 09:38:29

%S 1,1,0,21,182,2093,23394,285383,3621150,47813367,650302814,9066061193,

%T 128986881660,1866877313441,27417577343674,407771633368995,

%U 6131640430397790,93096368350684720,1425633583562073228,21998953427963954547,341803226976420690600

%N Number of unlabeled asymmetric 7-ary cacti having n polygons.

%H Andrew Howroyd, <a href="/A054370/b054370.txt">Table of n, a(n) for n = 0..200</a>

%H Miklos Bona, Michel Bousquet, Gilbert Labelle, and Pierre Leroux, <a href="https://doi.org/10.1006/aama.1999.0665">Enumeration of m-ary cacti</a>, Advances in Applied Mathematics, 24 (2000), 22-56.

%H <a href="/index/Ca#cacti">Index entries for sequences related to cacti</a>

%F a(n) = (1/n)*(Sum_{d|n} mu(n/d)*binomial(7*d, d)) - 6*binomial(7*n, n)/(6*n+1) for n > 0. - _Andrew Howroyd_, May 02 2018

%t a[0] = 1;

%t a[n_] := DivisorSum[n, MoebiusMu[n/#] Binomial[7#, #]&]/n - 6 Binomial[7n, n]/(6n + 1);

%t Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Jul 01 2018, after _Andrew Howroyd_ *)

%o (PARI) a(n) = if(n==0, 1, sumdiv(n, d, moebius(n/d)*binomial(7*d, d))/n - 6*binomial(7*n, n)/(6*n+1)) \\ _Andrew Howroyd_, May 02 2018

%Y Column k=7 of A303913.

%Y Cf. A054369, A054371.

%K nonn

%O 0,4

%A _Simon Plouffe_

%E Terms a(11) and beyond from _Andrew Howroyd_, May 02 2018