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%I M2591 #34 Dec 16 2021 22:15:59
%S 0,1,1,1,3,6,16,43,120,339,985,2892,8606,25850,78347,239161,734922,
%T 2271085,7054235,22010418,68958139,216842102,684164551,2165240365,
%U 6871792256,21865189969,69737972975,222915760126,714001019626,2291298553660,7366035776888
%N Number of asymmetric rooted connected graphs where every block is a complete graph.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Alois P. Heinz, <a href="/A007561/b007561.txt">Table of n, a(n) for n = 0..1900</a>
%H M. Bernstein and N. J. A. Sloane, <a href="http://arXiv.org/abs/math.CO/0205301">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
%H M. Bernstein and N. J. A. Sloane, <a href="/A003633/a003633_1.pdf">Some canonical sequences of integers</a>, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F Shifts left when weigh-transform applied twice.
%F a(n) ~ c * d^n / n^(3/2), where d = 3.382016466020272807429818743..., c = 0.161800727760188847021075748... . - _Vaclav Kotesovec_, Jul 26 2014
%p g:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p add(binomial(a(i), j)*g(n-i*j, i-1), j=0..n/i)))
%p end:
%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p add(binomial(g(i, i), j)*b(n-i*j, i-1), j=0..n/i)))
%p end:
%p a:= n-> `if`(n<1, 0, b(n-1, n-1)):
%p seq(a(n), n=0..40); # _Alois P. Heinz_, May 19 2013
%t g[n_, i_] := g[n, i] = If[n == 0, 1, If[i<1, 0, Sum[Binomial[a[i], j]*g[n-i*j, i-1], {j, 0, n/i}]]]; b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[Binomial[g[i, i], j]*b[n-i*j, i-1], {j, 0, n/i}]]]; a[n_] := If[n<1, 0, b[n-1, n-1]]; Table[a[n] // FullSimplify, {n, 0, 40}] (* _Jean-François Alcover_, Feb 11 2014, after _Alois P. Heinz_ *)
%Y Cf. A007563, A035079-A035081.
%Y Column k=2 of A316101.
%K nonn,nice,eigen
%O 0,5
%A _N. J. A. Sloane_
%E Additional comments from _Christian G. Bower_
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