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.

Previous Showing 11-18 of 18 results.

A002991 Number of n-node trees with a forbidden limb of length 5.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 5, 10, 21, 43, 97, 215, 503, 1187, 2876, 7033, 17510, 43961, 111664, 285809, 737632, 1915993, 5008652, 13163785, 34774873, 92282214, 245930746, 657931603, 1766481135, 4758553683, 12858286083, 34844908142, 94681272368
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. - Christian G. Bower, Dec 15 1999

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-A002992, A052318-A052329.

Programs

  • Maple
    with(numtheory):
g:= proc(n) g(n):= `if`(n=0, 1, add(add(d*(g(d-1)-
      `if`(d=5, 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 == 5, 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 A052328. - Christian G. Bower, Dec 15 1999
a(n) ~ c * d^n / n^(5/2), where d = 2.9447916575019743775137795109303..., c = 0.521642401804532770865780146005... . - Vaclav Kotesovec, Aug 25 2014

Extensions

More terms from Christian G. Bower, Dec 15 1999

A052324 Number of increasing rooted trees with a forbidden limb of length 3.

Original entry on oeis.org

0, 1, 1, 2, 5, 19, 90, 520, 3475, 26550, 228050, 2177020, 22860090, 261870070, 3249793360, 43432062300, 621911561150, 9498946124800, 154152712434600, 2648808048264400, 48043086765929200, 917249983543337400
Offset: 0

Views

Author

Christian G. Bower, Dec 15 1999

Keywords

Comments

In an increasing rooted tree, nodes are numbered and numbers increase as you move away from root.
A rooted tree with a forbidden limb of length k is a rooted tree where the path from any leaf inward hits a branching node or the root within k steps.

Links

Crossrefs

Cf. A002955, A002988-A002992, A052319-A052329.

Programs

  • Mathematica
    CoefficientList[Assuming[{Element[x, Reals], x > 0}, Series[-Log[1-6^(1/3)*Gamma[1/3]/3 + 1/3*x*ExpIntegralE[2/3, x^3/6]], {x, 0, 20}]], x]*Range[0, 20]! (* Vaclav Kotesovec, Mar 28 2014 *)
  • PARI
    {a(n)=local(A=x); for(i=0, n, A=intformal(exp(A-x^3/6+O(x^n)) )); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", ")) \\ Vaclav Kotesovec, Mar 28 2014

Formula

E.g.f. satisfies A'(x) = exp(A(x) - x^3/6). - corrected by Vaclav Kotesovec, Mar 28 2014
a(n) ~ d^n * (n-1)!, where d = 0.9546118344740519430556804... - Vaclav Kotesovec, Mar 28 2014
In closed form, d = 1/r, where r = 1.04754620033697244977759528695194261... is the root of the equation 1 = Integral_{x=0..r} exp(-x^3/6) dx. - Vaclav Kotesovec, Aug 21 2014

A052325 Number of asymmetric rooted trees with a forbidden limb of length 3.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 4, 8, 15, 30, 60, 122, 249, 513, 1061, 2210, 4620, 9708, 20472, 43337, 92023, 196018, 418653, 896485, 1924154, 4139014, 8921349, 19266067, 41679483, 90318082, 196020800, 426055601, 927317334, 2020949226, 4409764169
Offset: 1

Views

Author

Christian G. Bower, Dec 15 1999

Keywords

Comments

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

Links

Crossrefs

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

Programs

  • Maple
    b:= proc(n, i) option remember;
      `if`(n=0, 1, `if`(i<1, 0, add(binomial(a(i)-
      `if`(i=3, 1, 0), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> `if`(n<1, 1, b(n-1, n-1)):
seq(a(n), n=1..50);  # Alois P. Heinz, Jul 06 2014
  • Mathematica
    b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, Sum[Binomial[a[i]- If[i==3, 1, 0], j]*b[n-i*j, i-1], {j, 0, n/i}]]];
a[n_] := If[n<1, 1, b[n-1, n-1]];
Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Mar 01 2016, after Alois P. Heinz *)

Formula

a(n) satisfies a = SHIFT_RIGHT(WEIGH(a-b)) where b(3)=1, b(k)=0 if k != 3.
a(n) ~ c * d^n / n^(3/2), where d = 2.27671458388797627098091744865..., c = 0.2935911773459468433271794078... . - Vaclav Kotesovec, Aug 25 2014

A052327 Number of rooted trees with a forbidden limb of length 4.

Original entry on oeis.org

1, 1, 2, 4, 8, 18, 43, 102, 251, 625, 1584, 4055, 10509, 27451, 72307, 191697, 511335, 1370995, 3693452, 9991671, 27133149, 73934800, 202096673, 553999573, 1522651908, 4195087022, 11583820212, 32052475655, 88860186023
Offset: 1

Views

Author

Christian G. Bower, Dec 15 1999

Keywords

Comments

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

Links

Crossrefs

Cf. A002955, A002988-A002992, A052318-A052329.
Column k=4 of A255636.

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-> g(n-1):
seq(a(n), n=1..35);  # Alois P. Heinz, Jul 04 2014
  • Mathematica
    g[n_] := g[n] = If[n == 0, 1, Sum[DivisorSum[j, #*(g[# - 1] - If[# == 4, 1, 0])&] * g[n - j], {j, 1, n}]/n];
a[n_] := g[n - 1];
Table[a[n], {n, 1, 35}] (* Jean-François Alcover, Apr 04 2017, after Alois P. Heinz *)

Formula

a(n) satisfies a = SHIFT_RIGHT(EULER(a-b)) where b(4)=1, b(k)=0 if k != 4.
a(n) ~ c * d^n / n^(3/2), where d = 2.9224691962496551739365155005926289..., c = 0.43112017460637374030857983498164... . - Vaclav Kotesovec, Aug 25 2014

A052320 Number of labeled trimmed trees with n nodes.

Original entry on oeis.org

1, 1, 1, 0, 4, 5, 96, 1057, 14848, 235881, 4234240, 84815621, 1877090304, 45524670061, 1201345331200, 34283233751145, 1052350187831296, 34582597023733073, 1211614017182760960, 45088964565749965837
Offset: 0

Views

Author

Christian G. Bower, Dec 15 1999

Keywords

Comments

A trimmed tree is a tree with a forbidden limb of length 2.
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^3/2+x^4/2-LambertW[-x/E^(x^2)]-(LambertW[-x/E^(x^2)])^2/2, {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Mar 29 2014 *)

Formula

E.g.f.: 1 + x^3/2 + x^4/2 + B(x) - B(x)^2/2 where B(x) is e.g.f. of A052318.
a(n) ~ (1+LambertW(-2*exp(-2)))^(3/2) * exp((n*LambertW(-2*exp(-2)))/2) * n^(n-2). - Vaclav Kotesovec, Mar 29 2014

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

Views

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

A052326 Number of asymmetric trees with a forbidden limb of length 3.

Original entry on oeis.org

1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 6, 11, 25, 49, 105, 211, 444, 903, 1880, 3865, 8042, 16658, 34764, 72484, 151856, 318418, 669934, 1411637, 2982407, 6311760, 13387127, 28442458, 60543586, 129084965, 275683061, 589660911, 1263128375
Offset: 0

Views

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.

Formula

G.f.: 1 + B(x) + x^4 + x^5 + x^6 - (B(x^2) + B(x)^2)/2 where B(x) is g.f. of A052325.
a(n) ~ c * d^n / n^(5/2), where d = 2.2767145838879762709809174486..., c = 0.15900430026983804503695298... . - Vaclav Kotesovec, Aug 25 2014

Extensions

More terms, formula and comments from Christian G. Bower, Dec 15 1999
Typo in cross-reference corrected by Vaclav Kotesovec, Aug 25 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
Previous Showing 11-18 of 18 results.

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