cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A004114 Number of trees with n nodes and 2-colored internal (non-leaf) nodes.

Original entry on oeis.org

1, 1, 1, 2, 5, 12, 33, 98, 305, 1002, 3424, 12016, 43230, 158516, 590621, 2230450, 8521967, 32889238, 128064009, 502590642, 1986357307, 7900377892, 31602819524, 127076645038, 513419837168, 2083414420394, 8488377206876, 34712566540014, 142443837953632
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A004113, A052316, A052317.

Programs

  • Mathematica
    max = 28; 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}]/n ]; b]; bb = etr[A004113]; A004113[n_] := If[n <= 1, n, 2*bb[n - 1]]; b[x_] := Sum[A004113[n] x^n, {n, 1, max}]; f[x_] := Sum[a[n] x^n, {n, 0, max}]; a[0] = a[1] = a[2] = 1; coes = CoefficientList[ Series[f[x] - (1 + b[x] - x*b[x] - b[x]^2/2 + b[x^2]/2), {x, 0, max}], x]; Table[a[n], {n, 0, max}] /. Solve[Thread[coes == 0]][[1]] (* Jean-François Alcover, Jan 29 2013, after Alois P. Heinz *)

Formula

G.f.: 1+B(x)-x*B(x)-B(x)^2/2+B(x^2)/2 where B(x) is g.f. of A004113. - Christian G. Bower, Dec 15 1999
a(n) ~ c * d^n / n^(5/2), where d = 4.49415643203339504537343052... (same as for A004113), c = 0.31497820931312537077... . - Vaclav Kotesovec, Sep 12 2014

Extensions

More terms, and new description from Christian G. Bower, Dec 15 1999

A052316 Number of labeled rooted trees with n nodes and 2-colored internal (non-leaf) nodes.

Original entry on oeis.org

1, 4, 30, 344, 5370, 106452, 2562182, 72592816, 2367054450, 87320153900, 3595646533182, 163492924997448, 8136172620013802, 439858024910227588, 25670670464821310070, 1608575860476990991712, 107716675117341985862370
Offset: 1

Views

Author

Christian G. Bower, Dec 15 1999

Keywords

Links

Crossrefs

Cf. A004113, A004114, A052317.

Programs

  • Mathematica
    a[n_] := Sum[j^(n-1)*2^j*(-1)^(n-j)*Binomial[n, j], {j, 1, n}]; a[1] = 1; Table[a[n], {n, 1, 17}] (* Jean-François Alcover, Feb 26 2013, after Vladimir Kruchinin *)
  • Maxima
    a(n):=if n=1 then 1 else sum(j^(n-1)*2^j*(-1)^(n-j)*binomial(n,j),j,1,n); /* Vladimir Kruchinin, Jan 24 2012 */

Formula

Divides by 2n and shifts left under exponential transform.
E.g.f.: -x-LambertW(-2*x*exp(-x)). - Vladeta Jovovic, Sep 17 2003
a(n) = sum(j=1..n, j^(n-1)*2^j*(-1)^(n-j)*binomial(n,j)), n>1, a(1)=1. - Vladimir Kruchinin, Jan 24 2012
a(n) ~ sqrt(1+LambertW(-exp(-1)/2)) * n^(n-1) / (exp(n)*(-LambertW(-exp(-1)/2))^n). - Vaclav Kotesovec, Oct 05 2013

A052317 Number of labeled trees with n nodes and 2-colored internal (non-leaf) nodes.

Original entry on oeis.org

1, 1, 1, 6, 56, 730, 12372, 259574, 6511920, 190413234, 6364960940, 239556803662, 10028763883272, 462366507311306, 23282257730716740, 1271520006077859750, 74865320814990626912, 4727699146425478764898, 318763676354643090937692, 22856568223852002933212798
Offset: 0

Views

Author

Christian G. Bower, Dec 15 1999

Keywords

Links

Crossrefs

Cf. A004113, A004114, A052316.

Programs

  • Mathematica
    CoefficientList[Series[1+(1-x)*(-x-LambertW[-2*x*E^(-x)])-(-x-LambertW[-2*x*E^(-x)])^2/2, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Oct 05 2013 *)

Formula

E.g.f.: 1 + B(x) - x*B(x) - B(x)^2/2 where B(x) is e.g.f. of A052316.
a(n) ~ (1+LambertW(-exp(-1)/2))^(3/2) * n^(n-2) / (exp(n)*(-LambertW(-exp(-1)/2))^n). - Vaclav Kotesovec, Oct 05 2013

A108530 Number of rooted identity trees with n internal (non-leaf) nodes.

Original entry on oeis.org

1, 1, 2, 4, 12, 34, 110, 364, 1248, 4356, 15520, 56022, 204726, 755472, 2812004, 10543718, 39791070, 151022006, 576090250, 2207493080, 8493196536, 32797115398, 127071214442, 493831241234, 1924504466246, 7519182311366, 29447430754182, 115577336981932
Offset: 0

Views

Author

Christian G. Bower, Jun 07 2005

Keywords

Comments

Also for n>0, rooted trees with n nodes and 2-colored internal nodes. Black nodes correspond to nodes with a leaf child; white nodes correspond to those without one.

Links

Crossrefs

Cf. A004111, A004113, A108531, A108532.

Programs

  • Maple
    b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(a(i$2), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> `if`(n<2, 1, 2*b(n-1,n-1)):
seq(a(n), n=0..30);  # Alois P. Heinz, May 20 2013
  • Mathematica
    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, 1, 2*b[n-1, n-1]];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 01 2016, after Alois P. Heinz *)

Formula

Shifts left and halves under WEIGH transform.
a(n) ~ c * d^n / n^(3/2), where d = 4.1516890102085520777311008746639624... and c = 0.3329810927479684511418598248... - Vaclav Kotesovec, Feb 28 2014
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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