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A007560 Number of planted identity trees where non-root, non-leaf nodes an even distance from root are of degree 2.

%I A007560 M0325 #36 Dec 16 2021 22:15:45
%S A007560 1,1,1,1,2,2,4,6,10,17,29,51,89,159,284,512,930,1692,3101,5698,10515,
%T A007560 19464,36143,67296,125622,235050,440756,828142,1558955,2939761,
%U A007560 5552744,10504222,19899760,37750091,71704061,136361602,259618770,494821629,944074665
%N A007560 Number of planted identity trees where non-root, non-leaf nodes an even distance from root are of degree 2.
%D A007560 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007560 Alois P. Heinz, <a href="/A007560/b007560.txt">Table of n, a(n) for n = 1..1000</a>
%H A007560 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 A007560 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 A007560 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H A007560 <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>
%F A007560 Shifts 2 places left under weigh transform.
%F A007560 a(n) ~ c * d^n / n^(3/2), d = 1.983229991815043367273184141..., c = 0.5857451140002020594085469... . - _Vaclav Kotesovec_, Aug 25 2014
%F A007560 G.f.: x + x^2 * Product_{n>=1} (1 + x^n)^a(n). - _Ilya Gutkovskiy_, May 09 2019
%p A007560 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A007560       add(binomial(a(i), j)*b(n-i*j, i-1), j=0..n/i)))
%p A007560     end:
%p A007560 a:= n-> `if`(n<2, n, b(n-2, n-2)):
%p A007560 seq(a(n), n=1..50);  # _Alois P. Heinz_, May 19 2013
%t A007560 b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[a[i], j]*b[n-i*j, i-1], {j, 0, n/i}]]]; a[n_] := If[n<2, n, b[n-2, n-2]]; Table[a[n], {n, 1, 40}] (* _Jean-François Alcover_, Jan 27 2014, after _Alois P. Heinz_ *)
%Y A007560 Cf. A007562.
%Y A007560 Column k=2 of A316074.
%K A007560 nonn,nice,eigen
%O A007560 1,5
%A A007560 _N. J. A. Sloane_
%E A007560 Better description from _Christian G. Bower_, May 15 1998