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A213682 Number of rooted trees with n nodes having some subtrees replaced by cycles such that no leaf nodes are left over. 3
0, 0, 0, 1, 2, 3, 4, 6, 9, 16, 28, 51, 91, 169, 312, 588, 1113, 2125, 4068, 7844, 15174, 29505, 57553, 112680, 221224, 435637, 859958, 1701775, 3374860, 6706722, 13352832, 26632446, 53205674, 106458153, 213317105, 428021186, 859924107, 1729741358, 3483357290 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

FORMULA

a(n) ~ c * d^n / n^(3/2), where d = 2.097576205396085315047533452..., c = 0.47373451480298310710830384... . - Vaclav Kotesovec, Sep 07 2014

EXAMPLE

..............................................

:   o   :   o      o   :   o      o      o   :

:  / \  :  / \     |   :  / \     |      |   :

: o---o : o   o    o   : o   o    o      o   :

:       :  \ /    / \  : |   |   / \     |   :

:       :   o    o---o : o---o  o   o    o   :

:       :              :         \ /    / \  :

: n=3   : n=4          : n=5      o    o---o :

:.......:..............:.....................:

:   o      o      o      o   :

:  / \     |      |      |   :

: o   o    o      o      o   :

: |   |   / \     |      |   :

: o   o  o   o    o      o   :

:  \ /   |   |   / \     |   :

:   o    o---o  o   o    o   :

:                \ /    / \  :

: n=6             o    o---o :

:............................:....................

:   o      o      o      o      o         o      :

:  / \     |      |      |      |        / \     :

: o   o    o      o      o      o       /   \    :

: |   |   / \     |      |      |      o     o   :

: o   o  o   o    o      o      o     / \   / \  :

: |   |  |   |   / \     |      |    o---o o---o :

: o---o  o   o  o   o    o      o                :

:         \ /   |   |   / \     |                :

:          o    o---o  o   o    o                :

:                       \ /    / \               :

: n=7                    o    o---o              :

:................................................:

MAPLE

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

      add(binomial(a(i)+j-1, j)*b(n-i*j, i-1), j=0..n/i)))

    end:

a:= n-> `if`(n<3, 0, 1+b(n-1, n-1)):

seq(a(n), n=0..40);

MATHEMATICA

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[Binomial[a[i]+j-1, j]*b[n-i*j, i-1], {j, 0, n/i}]] // FullSimplify]; a[n_] := If[n<3, 0, 1+b[n-1, n-1]]; Table[a[n], {n, 0, 40}] (* Jean-Fran├žois Alcover, Feb 05 2015, after Alois P. Heinz *)

CROSSREFS

Cf. A000081, A213674, A213683.

Sequence in context: A192267 A331022 A256774 * A103481 A215285 A275173

Adjacent sequences:  A213679 A213680 A213681 * A213683 A213684 A213685

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Mar 04 2013

STATUS

approved

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Last modified October 20 10:21 EDT 2021. Contains 348100 sequences. (Running on oeis4.)