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.

A303023 Number of anti-binary (no binary branchings) unlabeled rooted trees with n nodes.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 16, 32, 66, 139, 297, 642, 1404, 3097, 6888, 15428, 34770, 78785, 179397, 410264, 941935, 2170275, 5016604, 11630024, 27034824, 63000261, 147148341, 344419767, 807746487, 1897829065, 4466643367, 10529301944, 24858143953, 58769113863
Offset: 1

Views

Author

Gus Wiseman, Aug 15 2018

Keywords

Examples

			The a(6) = 8 rooted trees:
  (((((o)))))
  (((ooo)))
  ((oo(o)))
  (oo((o)))
  (o(o)(o))
  ((oooo))
  (ooo(o))
  (ooooo)

Links

Crossrefs

Cf. A000081, A000598, A001190, A001678, A102403, A292050, A298204, A298422.
Cf. A303022, A303024, A303025, A303026, A303027, A318520.

Programs

  • Maple
    b:= proc(n, i, t) option remember; `if`(n=0, `if`(t=1, 0, 1), `if`(i<1, 0,
      add(b(n-i*j, i-1, max(0, t-j))*binomial(a(i)+j-1, j), j=0..n/i)))
    end:
a:= n-> `if`(n<2, n, b(n-1$2, 3)):
seq(a(n), n=1..50);  # Alois P. Heinz, Aug 27 2018
  • Mathematica
    burt[n_]:=burt[n]=If[n==1,{{}},Join@@Table[Union[Sort/@Tuples[burt/@c]],{c,Select[IntegerPartitions[n-1],Length[#]!=2&]}]];
Table[Length[burt[n]],{n,20}]
(* Second program: *)
b[n_, i_, t_] := b[n, i, t] = If[n == 0, If[t == 1, 0, 1], If[i < 1, 0, Sum[b[n-i*j, i-1, Max[0, t-j]]*Binomial[a[i]+j-1, j], {j, 0, n/i}]]];
a[n_] := If[n < 2, n, b[n-1, n-1, 3]];
Array[a, 50] (* Jean-François Alcover, May 16 2021, after Alois P. Heinz *)

Extensions

a(24)-a(34) from Alois P. Heinz, Aug 27 2018

A303022 Number of free pure symmetric multifunctions (with empty expressions allowed) with one atom, n positions, and no unitary parts (subexpressions of the form x[y]).

Original entry on oeis.org

1, 1, 1, 2, 5, 12, 27, 63, 152, 376, 939, 2371, 6047, 15577, 40429, 105637, 277625, 733518, 1947126, 5190503, 13888811, 37291968, 100444019, 271316998, 734802247, 1994873116, 5427893149, 14799525982, 40429761365, 110645688034, 303316712450, 832799212777
Offset: 1

Views

Author

Gus Wiseman, Aug 15 2018

Keywords

Comments

Also the number of orderless Mathematica expressions with one atom, n positions, and no unitary parts.

Examples

			The a(6) = 12 Mathematica expressions:
  o[o,o[][]]
  o[o[],o[]]
  o[o,o,o[]]
  o[o,o,o,o]
  o[][o,o[]]
  o[][o,o,o]
  o[][][o,o]
  o[o,o[]][]
  o[o,o,o][]
  o[][o,o][]
  o[o,o][][]
  o[][][][][]

Links

Crossrefs

Cf. A000108, A001003, A001006, A007853, A102403, A126120, A318049.
Cf. A303023, A303024, A303025, A303026, A303027.

Programs

  • Mathematica
    allOLBF[n_]:=allOLBF[n]=If[n==1,{"o"},Join@@Cases[Table[PR[k,n-k-1],{k,n-1}],PR[h_,g_]:>Join@@Table[Apply@@@Tuples[{allOLBF[h],Select[Union[Sort/@Tuples[allOLBF/@p]],Length[#]!=1&]}],{p,IntegerPartitions[g]}]]];
Table[Length[allOLBF[n]],{n,10}]
  • PARI
    EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(v=[1]); for(n=2, n, my(t=EulerT(v)-v); v=concat(v, v[n-1] + sum(k=1, n-2, v[k]*t[n-k-1]))); v} \\ Andrew Howroyd, Aug 19 2018

Extensions

Terms a(21) and beyond from Andrew Howroyd, Aug 19 2018

A303024 Matula-Goebel numbers of anti-binary (no binary branchings) rooted trees.

Original entry on oeis.org

1, 2, 3, 5, 8, 11, 12, 16, 18, 19, 20, 24, 27, 30, 31, 32, 36, 37, 40, 44, 45, 48, 50, 53, 54, 60, 61, 64, 66, 67, 71, 72, 75, 76, 80, 81, 88, 89, 90, 96, 99, 100, 103, 108, 110, 113, 114, 120, 124, 125, 127, 128, 131, 132, 135, 144, 148, 150, 151, 152, 157
Offset: 1

Views

Author

Gus Wiseman, Aug 15 2018

Keywords

Examples

			The sequence of anti-binary rooted trees together with their Matula-Goebel numbers begins:
   1: o
   2: (o)
   3: ((o))
   5: (((o)))
   8: (ooo)
  11: ((((o))))
  12: (oo(o))
  16: (oooo)
  18: (o(o)(o))
  19: ((ooo))
  20: (oo((o)))
  24: (ooo(o))
  27: ((o)(o)(o))
  30: (o(o)((o)))
  31: (((((o)))))

Crossrefs

Cf. A000081, A000598, A001190, A001678, A007097, A061775, A111299, A292050, A298422, A298424.
Cf. A303022, A303023, A303025, A303026, A303027.

Programs

  • Mathematica
    abQ[n_]:=Or[n==1,And[PrimeOmega[n]!=2,And@@Cases[FactorInteger[n],{p_,_}:>abQ[PrimePi[p]]]]]
Select[Range[100],abQ]

A303025 Number of series-reduced anti-binary (no unary or binary branchings) unlabeled rooted trees with n nodes.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 2, 3, 4, 7, 11, 17, 28, 46, 74, 123, 205, 341, 571, 964, 1629, 2764, 4707, 8040, 13766, 23639, 40681, 70163, 121256, 209960, 364168, 632694, 1100906, 1918375, 3347346, 5848271, 10229977, 17915018, 31407088, 55116661, 96818589, 170229939
Offset: 1

Views

Author

Gus Wiseman, Aug 15 2018

Keywords

Examples

			The a(10) = 7 rooted trees:
  (oo(oo(ooo)))
  (o(ooo)(ooo))
  (oo(oooooo))
  (ooo(ooooo))
  (oooo(oooo))
  (ooooo(ooo))
  (ooooooooo)

Links

Crossrefs

Cf. A000081, A000598, A001190, A001678, A102403, A298204, A298422.
Cf. A303022, A303023, A303024, A303026, A303027.

Programs

  • Maple
    b:= proc(n, i, t) option remember; `if`(n=0, `if`(t=0, 1, 0), `if`(i<1, 0,
      add(b(n-i*j, i-1, max(0, t-j))*binomial(a(i)+j-1, j), j=0..n/i)))
    end:
a:= n-> `if`(n<2, n, b(n-1$2, 3)):
seq(a(n), n=1..50);  # Alois P. Heinz, Aug 27 2018
  • Mathematica
    zurt[n_]:=zurt[n]=If[n==1,{{}},Join@@Table[Union[Sort/@Tuples[zurt/@c]],{c,Select[IntegerPartitions[n-1],Length[#]>2&]}]];
Table[Length[zurt[n]],{n,20}]
(* Second program: *)
b[n_, i_, t_] := b[n, i, t] = If[n == 0, If[t == 0, 1, 0], If[i < 1, 0, Sum[b[n-i*j, i - 1, Max[0, t-j]]*Binomial[a[i]+j-1, j], {j, 0, n/i}]]];
a[n_] :=  If[n < 2, n, b[n-1, n-1, 3]];
Array[a, 50] (* Jean-François Alcover, May 17 2021, after Alois P. Heinz *)

Extensions

a(36)-a(42) from Alois P. Heinz, Aug 27 2018

A303026 Matula-Goebel numbers of series-reduced anti-binary (no unary or binary branchings) rooted trees.

Original entry on oeis.org

1, 8, 16, 32, 64, 76, 128, 152, 212, 256, 304, 424, 512, 524, 608, 722, 848, 1024, 1048, 1216, 1244, 1444, 1532, 1696, 2014, 2048, 2096, 2432, 2488, 2876, 2888, 3064, 3392, 3524, 4028, 4096, 4192, 4864, 4976, 4978, 5204, 5618, 5752, 5776, 6128, 6476, 6784
Offset: 1

Views

Author

Gus Wiseman, Aug 15 2018

Keywords

Examples

			The sequence of series-reduced anti-binary rooted trees together with their Matula-Goebel numbers begins:
     1: o
     8: (ooo)
    16: (oooo)
    32: (ooooo)
    64: (oooooo)
    76: (oo(ooo))
   128: (ooooooo)
   152: (ooo(ooo))
   212: (oo(oooo))
   256: (oooooooo)
   304: (oooo(ooo))
   424: (ooo(oooo))
   512: (ooooooooo)
   524: (oo(ooooo))
   608: (ooooo(ooo))
   722: (o(ooo)(ooo))
   848: (oooo(oooo))
  1024: (oooooooooo)
  1048: (ooo(ooooo))
  1216: (oooooo(ooo))
  1244: (oo(oooooo))
  1444: (oo(ooo)(ooo))
  1532: (oo(oo(ooo)))
  1696: (ooooo(oooo))
  2014: (o(ooo)(oooo))
  2048: (ooooooooooo)

Crossrefs

Cf. A000081, A000598, A001190, A001678, A007097, A061775, A102403, A111299, A292050, A298422, A298424.
Cf. A303022, A303023, A303024, A303025, A303027.

Programs

  • Mathematica
    azQ[n_]:=Or[n==1,And[PrimeOmega[n]>2,And@@Cases[FactorInteger[n],{p_,_}:>azQ[PrimePi[p]]]]]
Select[Range[1000],azQ]
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.