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%I #20 Jan 09 2024 08:48:18
%S 1,1,7,28,231,2100,23884,285390,3626295,47813815,650367788,9066061200,
%T 128987761308,1866877313448,27417589615234,407771633434368,
%U 6131640607962135,93096368350684727,1425633586192690945,21998953427963954554,341803227016091180620
%N Number of unlabeled 7-ary cacti having n polygons.
%H Andrew Howroyd, <a href="/A054369/b054369.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} phi(n/d)*binomial(7*d, d)) - 6*binomial(7*n, n)/(6*n+1) for n > 0. - _Andrew Howroyd_, May 02 2018
%F a(n) ~ 7^(7*n + 1/2) / (2 * sqrt(3*Pi) * n^(5/2) * 6^(6*n + 1)). - _Vaclav Kotesovec_, Jul 17 2017
%t a[n_] := If[n == 0, 1, (Binomial[7*n, n]/(6 n + 1) + DivisorSum[n, Binomial[7*#, #]*EulerPhi[n/#]*Boole[# < n] & ])/n]; Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Jul 17 2017 *)
%o (PARI) a(n) = if(n==0, 1, sumdiv(n, d, eulerphi(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 A303912.
%Y Cf. A054370, A054371.
%K nonn
%O 0,3
%A _Simon Plouffe_
%E More terms from _Jean-François Alcover_, Jul 17 2017
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