|
%I M2977 #45 Oct 30 2023 00:12:23
%S 1,1,3,14,89,716,6967,79524,1041541,15393100,253377811,4596600004,
%T 91112351537,1959073928124,45414287553455,1129046241331316,
%U 29965290866974493,845605519848379436,25282324544244718411,798348403914242674980,26549922456617388029641
%N Number of increasing rooted connected graphs where every block is a complete graph.
%C In an increasing rooted graph, nodes are numbered and the numbers increase as you move away from the root.
%C (a(n+1)/a(n))/n tends to 1/A073003 = 1.676875... (same limit as A029768). - _Vaclav Kotesovec_, Jul 26 2014
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Vaclav Kotesovec, <a href="/A007549/b007549.txt">Table of n, a(n) for n = 1..410</a> (first 200 terms from Vincenzo Librandi)
%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]
%F Shifts left when exponentiated twice.
%p exptr:= proc(p) local g; g:= proc(n) option remember; p(n) +add(binomial(n-1, k-1) *p(k) *g(n-k), k=1..n-1) end: end: b:= exptr(exptr(a)): a:= n-> `if`(n=0, 1, b(n-1)): seq(a(n), n=1..30); # _Alois P. Heinz_, Oct 07 2008
%t exptr[p_] := Module[{g}, g[n_] := g[n] = p[n] + Sum[ Binomial[n-1, k-1]*p[k]*g[n-k], {k, 1, n-1}]; g]; b = exptr[ exptr[a] ]; a[n_] := If[n == 0, 1, b[n-1]]; Table[ a[n], {n, 1, 19}] (* _Jean-François Alcover_, May 10 2012, after _Alois P. Heinz_ *)
%Y Cf. A007563, A030019, A035051-A035053.
%Y Cf. A029768.
%K nonn,eigen,nice
%O 1,3
%A _N. J. A. Sloane_
%E New description from _Christian G. Bower_, Oct 15 1998
|
