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 A007562 Number of planted trees where non-root, non-leaf nodes an even distance from root are of degree 2. (Formerly M0773) 27
 1, 1, 1, 2, 3, 6, 10, 20, 36, 72, 137, 275, 541, 1098, 2208, 4521, 9240, 19084, 39451, 82113, 171240, 358794, 753460, 1587740, 3353192, 7100909, 15067924, 32044456, 68272854, 145730675, 311575140, 667221030, 1430892924, 3072925944, 6607832422, 14226665499 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS There is no planted tree on one node by definition. Column k=2 of A144018. - Alois P. Heinz, Oct 17 2012 It appears that a(n) is also the number of locally non-intersecting unlabeled rooted trees with n nodes, where a tree is locally non-intersecting if the branches directly under of any non-leaf node have empty intersection. - Gus Wiseman, Aug 22 2018 REFERENCES N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS Alois P. Heinz, Table of n, a(n) for n = 1..1000 M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, arXiv:math/0205301 [math.CO], 2002; Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version] M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures] N. J. A. Sloane, Transforms FORMULA Shifts left 2 places under Euler transform. G.f.: x + x^2 / (Product_{k>0} (1 - x^k)^a(k)). - Michael Somos, Oct 06 2003 a(n) ~ c * d^n / n^(3/2), where d = 2.246066877341161662499621547921... and c = 0.68490297576105466417608032... . - Vaclav Kotesovec, Jun 23 2014 G.f. A(x) satisfies: A(x) = x + x^2 * exp(A(x) + A(x^2)/2 + A(x^3)/3 + A(x^4)/4 + ...). - Ilya Gutkovskiy, Jun 11 2021 EXAMPLE G.f. = x + x^2 + x^3 + 2*x^4 + 3*x^5 + 6*x^6 + 10*x^7 + 20*x^8 + 36*x^9 + ... From Joerg Arndt, Jun 23 2014: (Start) The a(8) = 20 such trees have the following level sequences: 01: [ 0 1 2 3 4 3 2 1 ] 02: [ 0 1 2 3 3 3 2 1 ] 03: [ 0 1 2 3 3 2 2 1 ] 04: [ 0 1 2 3 3 2 1 1 ] 05: [ 0 1 2 3 2 3 2 1 ] 06: [ 0 1 2 3 2 2 2 1 ] 07: [ 0 1 2 3 2 2 1 1 ] 08: [ 0 1 2 3 2 1 2 1 ] 09: [ 0 1 2 3 2 1 1 1 ] 10: [ 0 1 2 2 2 2 2 1 ] 11: [ 0 1 2 2 2 2 1 1 ] 12: [ 0 1 2 2 2 1 2 1 ] 13: [ 0 1 2 2 2 1 1 1 ] 14: [ 0 1 2 2 1 2 2 1 ] 15: [ 0 1 2 2 1 2 1 1 ] 16: [ 0 1 2 2 1 1 1 1 ] 17: [ 0 1 2 1 2 1 2 1 ] 18: [ 0 1 2 1 2 1 1 1 ] 19: [ 0 1 2 1 1 1 1 1 ] 20: [ 0 1 1 1 1 1 1 1 ] Successive levels change by at most 1 and the last level is 1, compare to the example in A000081. (End) From Gus Wiseman, Aug 22 2018: (Start) The a(7) = 10 locally non-intersecting trees: (o(o(oo))) (o(oo(o))) (o(oooo)) (oo(o(o))) (oo(ooo)) (o(o)(oo)) (ooo(oo)) (oo(o)(o)) (oooo(o)) (oooooo) (End) MAPLE with(numtheory): etr:= proc(p) local b; b:= proc(n) option remember; if n=0 then 1 else (add(d*p(d), d=divisors(n)) +add(add(d*p(d), d= divisors(j)) *b(n-j), j=1..n-1))/n fi end end: b:= etr(a): a:= n-> `if`(n<=1, n, b(n-2)): seq(a(n), n=1..40); # Alois P. Heinz, Sep 06 2008 MATHEMATICA etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, (Sum[ Sum[ d*p[d], {d, Divisors[j]}]*b[n-j], {j, 1, n-1}] + Sum[ d*p[d], {d, Divisors[n]}])/n]; b]; b = etr[a]; a[n_] := If[n <= 1, n, b[n-2]]; Table[a[n], {n, 1, 36}] (* Jean-François Alcover, Aug 01 2013, after Alois P. Heinz *) purt[n_]:=If[n==1, {{}}, Join@@Table[Select[Union[Sort/@Tuples[purt/@ptn]], Intersection@@#=={}&], {ptn, IntegerPartitions[n-1]}]]; Table[Length[purt[n]], {n, 10}] (* Gus Wiseman, Aug 22 2018 *) PROG (PARI) {a(n) = local(A); if( n<2, n>0, A = x / (1 - x) + O(x^n); for(k=2, n-2, A /= (1 - x^k + O(x^n))^polcoeff(A, k-1)); polcoeff(A, n-1))}; /* Michael Somos, Oct 06 2003 */ CROSSREFS Cf. A000081, A000837, A004111, A007560, A144018, A289509, A316470, A316473, A316475. Sequence in context: A005418 A329699 A002215 * A345973 A329702 A222855 Adjacent sequences: A007559 A007560 A007561 * A007563 A007564 A007565 KEYWORD nonn,nice,eigen AUTHOR EXTENSIONS Better description from Christian G. Bower, May 15 1998 STATUS approved

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Last modified January 27 17:42 EST 2023. Contains 359845 sequences. (Running on oeis4.)