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 1-5 of 5 results.

A032200 Number of rooted compound windmills (mobiles) of n nodes.

Original entry on oeis.org

1, 1, 2, 4, 9, 20, 51, 128, 345, 940, 2632, 7450, 21434, 62174, 182146, 537369, 1596133, 4767379, 14312919, 43162856, 130695821, 397184252, 1211057426, 3703794849, 11358759346, 34923477315, 107627138308, 332404636811
Offset: 1

Views

Author

Christian G. Bower

Keywords

Comments

Also the number of locally necklace plane trees with n nodes, where a plane tree is locally necklace if the sequence of branches directly under any given node is lexicographically minimal among its cyclic permutations. - Gus Wiseman, Sep 05 2018

Examples

			From _Gus Wiseman_, Sep 05 2018: (Start)
The a(5) = 9 locally necklace plane trees:
  ((((o))))
  (((oo)))
  ((o(o)))
  (o((o)))
  ((o)(o))
  ((ooo))
  (o(oo))
  (oo(o))
  (oooo)
(End)

References

  • F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 241 (3.3.84).

Links

Crossrefs

Cf. A029768, A038037, A055340.
Cf. A000108, A007853, A032171, A254040, A304173, A304175, A317852.

Programs

  • Mathematica
    neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
neckplane[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[neckplane/@c],neckQ],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[neckplane[n]],{n,10}] (* Gus Wiseman, Sep 05 2018 *)
  • PARI
    CIK(p,n)={sum(d=1, n, eulerphi(d)/d*log(subst(1/(1+O(x*x^(n\d))-p), x, x^d)))}
seq(n)={my(p=O(1));for(i=1, n, p=1+CIK(x*p, i)); Vec(p)} \\ Andrew Howroyd, Jun 20 2018

Formula

Shifts left under "CIK" (necklace, indistinct, unlabeled) transform.

A032171 Number of rooted compound windmills (mobiles) of n nodes with no symmetries.

Original entry on oeis.org

1, 1, 1, 2, 4, 10, 23, 59, 148, 385, 1006, 2678, 7170, 19421, 52933, 145364, 401421, 1114713, 3109710, 8713076, 24506121, 69168705, 195849114, 556165311, 1583601840, 4520226558, 12931917204, 37075154703
Offset: 1

Views

Author

Christian G. Bower

Keywords

Comments

Also the number of locally Lyndon plane trees with n nodes, where a plane tree is locally Lyndon if the sequence of branches directly under any given node is a Lyndon word. - Gus Wiseman, Sep 05 2018

Examples

			From _Gus Wiseman_, Sep 05 2018: (Start)
The a(6) = 10 locally Lyndon plane trees:
  (((((o)))))
  (((o(o))))
  ((o((o))))
  (o(((o))))
  ((o)((o)))
  ((oo(o)))
  (o(o(o)))
  (oo((o)))
  (o(o)(o))
  (ooo(o))
(End)

Links

Crossrefs

Cf. A032200, A055363.
Cf. A000108, A007853, A032171, A254040, A304173, A304175, A317852.

Programs

  • Mathematica
    T[n_, k_] := Module[{A}, A[, ] = 0; If[k < 1 || k > n, 0, For[j = 1, j <= n, j++, A[x_, y_] = x*y - x*Sum[MoebiusMu[i]/i * Log[1 -  A [x^i, y^i]] + O[x]^j // Normal , {i, 1, j}]]; Coefficient[Coefficient[A[x, y], x, n], y, k]]];
a[n_] := a[n] = Sum[T[n, k], {k, 1, n}];
Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 1, 28}] (* Jean-François Alcover, Jun 30 2017, using Michael Somos' code for A055363 *)
LyndonQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And]&&Array[RotateRight[q,#]&,Length[q],1,UnsameQ];
lynplane[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[lynplane/@c],LyndonQ],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[lynplane[n]],{n,10}] (* Gus Wiseman, Sep 05 2018 *)
  • PARI
    CHK(p,n)={sum(d=1, n, moebius(d)/d*log(subst(1/(1+O(x*x^(n\d))-p), x, x^d)))}
seq(n)={my(p=O(1));for(i=1, n, p=1+CHK(x*p, i)); Vec(p)} \\ Andrew Howroyd, Jun 20 2018

Formula

Shifts left under "CHK" (necklace, identity, unlabeled) transform.
From Petros Hadjicostas, Dec 03 2017: (Start)
a(n+1) = (1/n)*Sum_{d|n} mu(n/d)*c(d), where c(n) = n*a(n) + Sum_{s=1..n-1} c(s)*a(n-s) with a(1) = c(1) = 1.
G.f.: If A(x) = Sum_{n>=1} a(n)*x^n, then Sum_{n>=1} a(n+1)*x^n = -Sum_{n>=1} (mu(n)/n)*log(1-A(x^n)).
The g.f. of the auxiliary sequence (c(n): n>=1) is C(x) = Sum_{n>=1} c(n)*x^n = x*(dA(x)/dx)/(1-A(x)) = x + 3*x^2 + 7*x^3 + 19*x^4 + 51*x^5 + 147*x^6 + 414*x^7 + 1203*x^8 + ...
(End)

A304175 Number of leaf-balanced rooted plane trees with n nodes.

Original entry on oeis.org

1, 1, 2, 5, 12, 27, 59, 128, 277, 597, 1280, 2730, 5794, 12248, 25836, 54508, 115222, 244144, 518104, 1099499, 2330326, 4930089, 10415135, 21992400, 46470911, 98353146, 208580686, 443186181, 942988423, 2007981801, 4276830431, 9109431322, 19404918449, 41357252072, 88236092543
Offset: 1

Views

Author

Gus Wiseman, Aug 16 2018

Keywords

Comments

A rooted plane tree is leaf-balanced if every branch of the root has the same number of leaves, and every branch of the root is itself leaf-balanced.

Examples

			The a(5) = 12 leaf-balanced plane trees:
  ((((o)))), (((oo))), (((o)o)), ((o(o))), ((ooo)),
  (((o))o), (o((o))), ((o)(o)),
  ((o)oo), (o(o)o), (oo(o)),
  (oooo).
Missing from this list are ((oo)o) and (o(oo)).

Links

Crossrefs

Cf. A000081, A000108, A001003, A001006, A003238, A007853, A126120, A291442, A291443, A304173.

Programs

  • Mathematica
    lbplane[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[lbplane/@c],SameQ@@(Count[#,{},{0,Infinity}]&/@#)&],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[lbplane[n]],{n,10}]
  • PARI
    seq(n)={my(v=vector(n)); v[1]=x/(1-x) + O(x*x^n); for(k=2, n, v[k]=x*sumdiv(k, d, if(dAndrew Howroyd, Dec 13 2020

Extensions

Terms a(17) and beyond from Andrew Howroyd, Dec 13 2020

A317852 Number of plane trees with n nodes where the sequence of branches directly under any given node is aperiodic, meaning its cyclic permutations are all different.

Original entry on oeis.org

1, 1, 1, 3, 8, 26, 76, 247, 783, 2565, 8447, 28256, 95168, 323720, 1108415, 3821144, 13246307, 46158480, 161574043, 567925140, 2003653016, 7092953340, 25186731980, 89690452750, 320221033370, 1146028762599, 4110596336036, 14774346783745, 53203889807764, 191934931634880
Offset: 1

Views

Author

Gus Wiseman, Sep 05 2018

Keywords

Comments

Also the number of plane trees with n nodes where the sequence of branches directly under any given node has relatively prime run-lengths.

Examples

			The a(5) = 8 locally aperiodic plane trees:
  ((((o)))),
  (((o)o)), ((o(o))), (((o))o), (o((o))),
  ((o)oo), (o(o)o), (oo(o)).
The a(6) = 26 locally aperiodic plane trees:
  (((((o)))))  ((((o)o)))  (((o)oo))  ((o)ooo)
               (((o(o))))  ((o(o)o))  (o(o)oo)
               ((((o))o))  ((oo(o)))  (oo(o)o)
               ((o((o))))  (((o)o)o)  (ooo(o))
               ((((o)))o)  ((o(o))o)
               (o(((o))))  (o((o)o))
               (((o))(o))  (o(o(o)))
               ((o)((o)))  (((o))oo)
                           (o((o))o)
                           (oo((o)))
                           ((o)(o)o)
                           ((o)o(o))
                           (o(o)(o))

Links

Crossrefs

Cf. A000108, A000837, A007853, A032171, A032200, A254040, A301700, A303386, A303431, A304173, A304175, A317708, A317852.

Programs

  • Mathematica
    aperQ[q_]:=Array[RotateRight[q,#]&,Length[q],1,UnsameQ];
aperplane[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[aperplane/@c],aperQ],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[aperplane[n]],{n,10}]
  • PARI
    Tfm(p, n)={sum(d=1, n, moebius(d)*(subst(1/(1+O(x*x^(n\d))-p), x, x^d)-1))}
seq(n)={my(p=O(1)); for(i=1, n, p=1+Tfm(x*p, i)); Vec(p)} \\ Andrew Howroyd, Feb 08 2020

Extensions

a(16)-a(17) from Robert Price, Sep 15 2018
Terms a(18) and beyond from Andrew Howroyd, Feb 08 2020

A319123 Number of series-reduced plane trees with n leaves such that each branch directly under any given node has a different number of leaves.

Original entry on oeis.org

1, 1, 3, 7, 21, 75, 277, 1083, 4419, 18493, 77729, 332557, 1444477, 6307225, 27912147, 123878207, 554733045, 2492087531, 11280537097, 51120499279, 233319480419, 1065835004917, 4895443823281, 22505853359485, 103958158302085, 480365303903637, 2229412587062123
Offset: 1

Views

Author

Gus Wiseman, Sep 11 2018

Keywords

Examples

			The a(4) = 7 plane trees:
  (oooo)
  (o(ooo))
  ((ooo)o)
  (o(o(oo)))
  (o((oo)o))
  ((o(oo))o)
  (((oo)o)o)

Crossrefs

Cf. A000108, A001003, A007853, A074206, A118376, A273873, A277130, A281113, A304173, A304175, A319122.

Programs

  • Mathematica
    b[n_]:=b[n]=1+Sum[Times@@b/@f,{f,Join@@Permutations/@Select[IntegerPartitions[n],And[Length[#]>1,UnsameQ@@#]&]}];
Array[b,30]
Showing 1-5 of 5 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.