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 1-3 of 3 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

A004113 Number of rooted trees with n nodes and 2-colored non-leaf nodes.

Original entry on oeis.org

1, 2, 6, 18, 60, 204, 734, 2694, 10162, 38982, 151920, 599244, 2389028, 9608668, 38945230, 158904230, 652178206, 2690598570, 11151718166, 46412717826, 193891596436, 812748036380, 3417407089470, 14410094628558, 60920843101858, 258169745573158, 1096494947168142
Offset: 1

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. A004114, A052316, A052317.

Programs

  • Maple
    with(numtheory): etr:= proc(p) local b; b:= proc(n) option remember; `if`(n=0, 1, (add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n))/n) end end: b:= etr(a): a:= n-> `if`(n<=1, n, 2*b(n-1)): seq(a(n), n=1..30); # 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}]/n ]; b]; b = etr[a]; a[n_] := If[n <= 1, n, 2*b[n - 1]]; Table[a[n], {n, 1, 27}] (* Jean-François Alcover, Jan 29 2013, translated from Alois P. Heinz's Maple program *)

Formula

Shifts left and halves under EULER transform.
a(n) ~ c * d^n / n^(3/2), where d = 4.49415643203339504537343052838796824... and c = 0.368722987377516657464802259... - Vaclav Kotesovec, Feb 28 2014

Extensions

Extended with better description from Christian G. Bower, Apr 15 1998

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
Showing 1-3 of 3 results.

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

Content is available under The OEIS End-User License Agreement.