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.

A052323 Number of labeled trees with a forbidden limb of length 3.

Original entry on oeis.org

1, 1, 1, 3, 4, 65, 576, 5887, 92464, 1680345, 34041520, 774906011, 19537590744, 540890740117, 16321259150392, 533305854910935, 18764822871806176, 707498057530634033, 28460428902580264416, 1216828054782241792435
Offset: 0

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Author

Christian G. Bower, Dec 15 1999

Keywords

Comments

A tree with a forbidden limb of length k is a tree where the path from any leaf inward hits a branching node or another leaf within k steps.

Links

Crossrefs

Cf. A002955, A002988-A002992, A052318-A052329.

Programs

  • Mathematica
    CoefficientList[Series[1+x^4/2+x^5/2+x^6/2-LambertW[-x*E^(-x^3)]-(LambertW[-x*E^(-x^3)])^2/2, {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Mar 29 2014 *)

Formula

E.g.f.: 1+x^4/2+x^5/2+x^6/2+B(x)-B(x)^2/2 where B(x) is e.g.f. of A052322.
a(n) ~ (1+LambertW(-3*exp(-3)))^(3/2) * exp(n/3*LambertW(-3*exp(-3))) * n^(n-2). - Vaclav Kotesovec, Mar 29 2014

A002990 Number of n-node trees with a forbidden limb of length 4.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 5, 9, 19, 38, 86, 188, 439, 1026, 2472, 5997, 14835, 36964, 93246, 236922, 607111, 1565478, 4062797, 10599853, 27797420, 73224806, 193709710, 514406793, 1370937140, 3665714528, 9831891555, 26445886506, 71325268179
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

A tree with a forbidden limb of length k is a tree where the path from any leaf inward hits a branching node or another leaf within k steps.

References

  • A. J. Schwenk, Almost all trees are cospectral, pp. 275-307 of F. Harary, editor, New Directions in the Theory of Graphs. Academic Press, NY, 1973.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A002955, A002988, A002989, A002990, A002991, A002992.
Cf. A052318, A052319, A052320.
Cf. A052321, A052322, A052323, A052324, A052325, A052326.
Cf. A052327, A052328, A052329.

Programs

  • Maple
    with(numtheory):
g:= proc(n) g(n):= `if`(n=0, 1, add(add(d*(g(d-1)-
      `if`(d=4, 1, 0)), d=divisors(j))*g(n-j), j=1..n)/n)
    end:
a:= n-> `if`(n=0, 1, g(n-1)+(`if`(irem(n, 2, 'r')=0,
         g(r-1), 0)-add(g(i-1)*g(n-i-1), i=1..n-1))/2):
seq(a(n), n=0..40);  # Alois P. Heinz, Jul 06 2014
  • Mathematica
    g[n_] := g[n] = If[n == 0, 1, Sum[Sum[d*(g[d-1]-If[d == 4, 1, 0]), {d, Divisors[j] }]*g[n-j], {j, 1, n}]/n]; a[n_] := If[n == 0, 1, g[n-1] + (If[Mod[n, 2 ] == 0, g[Quotient[n, 2]-1], 0] - Sum[g[i-1]*g[n-i-1], {i, 1, n-1}])/2]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 26 2015, after Alois P. Heinz *)

Formula

G.f.: 1 + B(x) + (B(x^2) - B(x)^2)/2 where B(x) is g.f. of A052327.
a(n) ~ c * d^n / n^(5/2), where d = 2.9224691962496551739365155005926..., c = 0.503471518908815272581177797536... . - Vaclav Kotesovec, Aug 25 2014

Extensions

More terms, formula and comments from Christian G. Bower, Dec 15 1999

A360433 E.g.f. satisfies A(x) = x * exp(A(x) + x^3).

Original entry on oeis.org

0, 1, 2, 9, 88, 865, 11016, 173929, 3227792, 69010785, 1670970160, 45198840841, 1350754588008, 44196732194641, 1571453132115608, 60331412278617705, 2487385479819549856, 109608035124514365121, 5140910415583354887648, 255708987797133857518345
Offset: 0

Views

Author

Seiichi Manyama, Feb 07 2023

Keywords

Links

Crossrefs

Cf. A216857, A360432.
Cf. A052322.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-lambertw(-x*exp(x^3)))))

Formula

E.g.f.: -LambertW( -x*exp(x^3) ).
a(n) ~ sqrt(1+LambertW(3*exp(-3))) * 3^(n/3) * n^(n-1) / (exp(n) * (LambertW(3*exp(-3)))^(n/3)). - Vaclav Kotesovec, Feb 07 2023
Showing 1-3 of 3 results.

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

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