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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)
3
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

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, 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

Index entries for sequences related to rooted trees

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

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(7) = 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)

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-Fran├žois Alcover, Aug 01 2013, after Alois P. Heinz *)

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

Sequence in context: A120421 A005418 A002215 * A222855 A171682 A066062

Adjacent sequences:  A007559 A007560 A007561 * A007563 A007564 A007565

KEYWORD

nonn,nice,eigen

AUTHOR

N. J. A. Sloane

EXTENSIONS

Better description from Christian G. Bower, May 15 1998

STATUS

approved

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Last modified February 20 20:12 EST 2018. Contains 299385 sequences. (Running on oeis4.)