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.

A244372 Number T(n,k) of unlabeled rooted trees with n nodes and maximal outdegree (branching factor) k; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 5, 2, 1, 0, 1, 10, 6, 2, 1, 0, 1, 22, 16, 6, 2, 1, 0, 1, 45, 43, 17, 6, 2, 1, 0, 1, 97, 113, 49, 17, 6, 2, 1, 0, 1, 206, 300, 136, 50, 17, 6, 2, 1, 0, 1, 450, 787, 386, 142, 50, 17, 6, 2, 1, 0, 1, 982, 2074, 1081, 409, 143, 50, 17, 6, 2, 1
Offset: 1

Views

Author

Joerg Arndt and Alois P. Heinz, Jun 26 2014

Keywords

Examples

			The A000081(5) = 9 rooted trees with 5 nodes sorted by maximal outdegree are:
:  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     :         :
:  |  :  / \  |     |                   :             :         :
:  o  : o   o o     o                   :             :         :
:  |  :                                 :             :         :
:  o  :                                 :             :         :
:     :                                 :             :         :
: -1- : ---------------2--------------- : -----3----- : ---4--- :
Thus row 5 = [0, 1, 5, 2, 1].
Triangle T(n,k) begins:
  1;
  0,  1;
  0,  1,   1;
  0,  1,   2,    1;
  0,  1,   5,    2,    1;
  0,  1,  10,    6,    2,   1;
  0,  1,  22,   16,    6,   2,   1;
  0,  1,  45,   43,   17,   6,   2,  1;
  0,  1,  97,  113,   49,  17,   6,  2,  1;
  0,  1, 206,  300,  136,  50,  17,  6,  2,  1;
  0,  1, 450,  787,  386, 142,  50, 17,  6,  2,  1;
  0,  1, 982, 2074, 1081, 409, 143, 50, 17,  6,  2,  1;

Links

Crossrefs

Columns k=2-10 give: A244398, A244399, A244400, A244401, A244402, A244403, A244404, A244405, A244406.
T(2n,n) gives A244407(n).
T(2n+1,n) gives A244410(n).
Row sum give A000081.
Cf. A244454.

Programs

  • Maple
    b:= proc(n, i, t, k) option remember; `if`(n=0, 1,
      `if`(i<1, 0, add(binomial(b((i-1)$2, k$2)+j-1, j)*
       b(n-i*j, i-1, t-j, k), j=0..min(t, n/i))))
    end:
T:= (n, k)-> b(n-1$2, k$2) -`if`(k=0, 0, b(n-1$2, k-1$2)):
seq(seq(T(n, k), k=0..n-1), n=1..14);
  • Mathematica
    b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[b[i-1, i-1, k, k]+j-1, j]*b[n-i*j, i-1, t-j, k], {j, 0, Min[t, n/i]}]]]; T[n_, k_] := b[n-1, n-1, k, k] - If[k == 0, 0, b[n-1, n-1, k-1, k-1]]; Table[Table[T[n, k], {k, 0, n-1}], {n, 1, 14}] // Flatten (* Jean-François Alcover, Jul 01 2014, translated from Maple *)

A299039 Number of rooted trees with 2n nodes where each node has at most n children.

Original entry on oeis.org

1, 1, 3, 17, 106, 693, 4690, 32754, 234746, 1719325, 12820920, 97039824, 743680508, 5759507657, 45006692668, 354425763797, 2809931206626, 22409524536076, 179655903886571, 1447023307374888, 11703779855021636, 95020085240320710, 774088021528328920
Offset: 0

Views

Author

Alois P. Heinz, Feb 01 2018

Keywords

Examples

			a(2) = 3:
   o     o       o
   |     |      / \
   o     o     o   o
   |    / \    |
   o   o   o   o
   |
   o

Links

Crossrefs

Cf. A051491, A100034, A187770, A244407, A299038, A299098.

Programs

  • Maple
    b:= proc(n, i, t, k) option remember; `if`(n=0, 1,
      `if`(i<1, 0, add(binomial(b((i-1)$2, k$2)+j-1, j)*
       b(n-i*j, i-1, t-j, k), j=0..min(t, n/i))))
    end:
a:= n-> `if`(n=0, 1, b(2*n-1$2, n$2)):
seq(a(n), n=0..25);
  • Mathematica
    b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, 1, If[i < 1, 0, Sum[ Binomial[b[i - 1, i - 1, k, k] + j - 1, j]*b[n - i*j, i - 1, t - j, k], {j, 0, Min[t, n/i]}]]];
a[n_] := If[n == 0, 1, b[2n - 1, 2n - 1, n, n]];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jun 04 2018, from Maple *)

Formula

a(n) = A299038(2n,n).
a(n) ~ c * d^n / n^(3/2), where d = A051491^2 = 8.736548423865419449938118272879... and c = A187770 / 2^(3/2) = 0.155536626247883986039760097126... - Vaclav Kotesovec, Feb 02 2018, updated Mar 17 2024

A244410 Number of unlabeled rooted trees with 2n+1 nodes and maximal outdegree (branching factor) n.

Original entry on oeis.org

1, 1, 5, 16, 49, 142, 415, 1198, 3473, 10048, 29118, 84376, 244747, 710198, 2062273, 5991417, 17416400, 50652247, 147384675, 429043389, 1249508946, 3640449678, 10610613551, 30937605075, 90237313082, 263288153073, 768449666116, 2243530461066, 6552016136666
Offset: 0

Views

Author

Joerg Arndt and Alois P. Heinz, Jun 27 2014

Keywords

Links

Crossrefs

Cf. A244372, A244407, A051491.

Programs

  • Maple
    b:= proc(n, i, t, k) option remember; `if`(n=0, 1,
      `if`(i<1, 0, add(binomial(b((i-1)$2, k$2)+j-1, j)*
       b(n-i*j, i-1, t-j, k), j=0..min(t, n/i))))
    end:
a:= n-> `if`(n=0, 1, b(2*n$2, n$2)-b(2*n$2, n-1$2)):
seq(a(n), n=0..30);
  • Mathematica
    b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[b[i - 1, i - 1, k, k] + j - 1, j]* b[n - i*j, i - 1, t - j, k], {j, 0, Min[t, n/i]}]] // FullSimplify] ; a[n_] := If[n == 0, 1, b[2*n, 2 n, n, n] - b[2*n, 2 n, n - 1, n - 1]]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 06 2015, after Maple *)

Formula

a(n) = A244372(2n+1,n).
a(n) ~ c * d^n / sqrt(n), where d = 2.955765285651994974714817524... is the Otter's rooted tree constant (see A051491), and c = 2.806733... . - Vaclav Kotesovec, Jul 11 2014
Showing 1-3 of 3 results.

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