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 1-10 of 12 results. Next

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 *)

A244530 Number T(n,k) of ordered unlabeled rooted trees with n nodes such that the minimal outdegree of inner nodes equals 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, 4, 0, 1, 0, 11, 2, 0, 1, 0, 36, 5, 0, 0, 1, 0, 117, 11, 3, 0, 0, 1, 0, 393, 28, 7, 0, 0, 0, 1, 0, 1339, 78, 8, 4, 0, 0, 0, 1, 0, 4630, 201, 21, 9, 0, 0, 0, 0, 1, 0, 16193, 532, 55, 10, 5, 0, 0, 0, 0, 1, 0, 57201, 1441, 121, 11, 11, 0, 0, 0, 0, 0, 1
Offset: 1

Views

Author

Joerg Arndt and Alois P. Heinz, Jun 29 2014

Keywords

Comments

T(1,0) = 1 by convention.

Examples

			T(5,1) = 11:
  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       o       o
  |  / \  |         | |         |
  o o   o o         o o         o
  |
  o
Triangle T(n,k) begins:
  1;
  0,     1;
  0,     1,   1;
  0,     4,   0,  1;
  0,    11,   2,  0,  1;
  0,    36,   5,  0,  0, 1;
  0,   117,  11,  3,  0, 0, 1;
  0,   393,  28,  7,  0, 0, 0, 1;
  0,  1339,  78,  8,  4, 0, 0, 0, 1;
  0,  4630, 201, 21,  9, 0, 0, 0, 0, 1;
  0, 16193, 532, 55, 10, 5, 0, 0, 0, 0, 1;

Links

Crossrefs

Columns k=0-10 give: A063524, A106640(n-2), A244531, A244532, A244533, A244534, A244535, A244536, A244537, A244538, A244539.
Row sums give A000108(n-1).
Cf. A244454 (unordered unlabeled rooted trees).

Programs

  • Maple
    b:= proc(n, t, k) option remember; `if`(n=0,
      `if`(t in [0, k], 1, 0), `if`(t>n, 0, add(b(j-1, k$2)*
       b(n-j, max(0, t-1), k), j=1..n)))
    end:
T:= (n, k)-> b(n-1, k$2) -`if`(n=1 and k=0, 0, b(n-1, k+1$2)):
seq(seq(T(n, k), k=0..n-1), n=1..14);
  • Mathematica
    b[n_, t_, k_] := b[n, t, k] = If[n == 0, If[t == 0 || t == k, 1, 0], If[t>n, 0, Sum[b[j-1, k, k]*b[n-j, Max[0, t-1], k], {j, 1, n}]]]; T[n_, k_] := b[n-1, k, k] - If[n == 1 && k == 0, 0, b[n-1, k+1, k+1]]; Table[Table[T[n, k], {k, 0, n-1}], {n, 1, 14}] // Flatten (* Jean-François Alcover, Jan 13 2015, translated from Maple *)

A244456 Number of unlabeled rooted trees with n nodes such that the minimal outdegree of inner nodes equals 2.

Original entry on oeis.org

1, 0, 1, 2, 4, 7, 15, 28, 56, 110, 220, 436, 878, 1762, 3560, 7205, 14650, 29838, 60991, 124938, 256628, 528238, 1089834, 2252860, 4666304, 9682422, 20125777, 41900433, 87369029, 182441944, 381499040, 798782945, 1674575394, 3514733683, 7385298837, 15534856067
Offset: 3

Views

Author

Joerg Arndt and Alois P. Heinz, Jun 29 2014

Keywords

Examples

			a(5) = 1:
      o
     / \
    o   o
   / \
  o   o

Links

Crossrefs

Column k=2 of A244454.
Cf. A244531, A246403.

Programs

  • Maple
    b:= proc(n, i, t, k) option remember; `if`(n=0, `if`(t in [0, k],
      1, 0), `if`(i<1 or t>n, 0, add(binomial(b((i-1)$2, k$2)+j-1, j)*
      b(n-i*j, i-1, max(0,t-j), k), j=0..n/i)))
    end:
a:= n-> b(n-1$2, 2$2) -b(n-1$2, 3$2):
seq(a(n), n=3..40);
  • Mathematica
    b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, If[t == 0 || t == k, 1, 0], If[i < 1, 0, Sum[Binomial[b[i - 1, i - 1, k, k] + j - 1, j]*b[n - i*j, i - 1, Max[0, t - j], k], {j, 0, n/i}]]]; a[n_] := b[n - 1, n - 1, 2, 2] - b[n - 1, n - 1, 3, 3] // FullSimplify; Table[a[n], {n, 3, 40}] (* Jean-François Alcover, Feb 06 2015, after Maple *)

Formula

a(n) ~ c * d^n / n^(3/2), where d = A246403 = 2.18946198566085056388702757711..., c = 0.4213018528699249210965028... (constants are same as for A001679). - Vaclav Kotesovec, Jul 02 2014

A244455 Number of unlabeled rooted trees with n nodes such that the minimal outdegree of inner nodes equals 1.

Original entry on oeis.org

1, 1, 3, 7, 17, 42, 105, 267, 684, 1775, 4639, 12238, 32491, 86859, 233496, 631082, 1713613, 4673455, 12795426, 35159212, 96927479, 268021520, 743188706, 2066071045, 5757360011, 16079027344, 44997313684, 126166307275, 354384737204, 997083779801, 2809751278062
Offset: 2

Views

Author

Joerg Arndt and Alois P. Heinz, Jun 29 2014

Keywords

Examples

			a(5) = 7:
  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

Column k=1 of A244454.
Cf. A106640 (the same for ordered rooted trees).

Programs

  • Maple
    b:= proc(n, i, t, k) option remember; `if`(n=0, `if`(t in [0, k],
      1, 0), `if`(i<1 or t>n, 0, add(binomial(b((i-1)$2, k$2)+j-1, j)*
      b(n-i*j, i-1, max(0,t-j), k), j=0..n/i)))
    end:
a:= n-> b(n-1$2, 1$2) -b(n-1$2, 2$2):
seq(a(n), n=2..35);
  • Mathematica
    b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, If[t == 0 || t == k, 1, 0], If[i < 1, 0, Sum[Binomial[b[i - 1, i - 1, k, k] + j - 1, j]*b[n - i*j, i - 1, Max[0, t - j], k], {j, 0, n/i}]] // FullSimplify]; a[n_] := b[n - 1, n - 1, 1, 1] - b[n - 1, n - 1, 2, 2]; Table[a[n], {n, 2, 35}] (* Jean-François Alcover, Feb 06 2015, after Maple *)

Formula

a(n) = A000081(n) - A001678(n+1).

A244457 Number of unlabeled rooted trees with n nodes such that the minimal outdegree of inner nodes equals 3.

Original entry on oeis.org

1, 0, 0, 1, 2, 2, 4, 7, 12, 20, 34, 56, 98, 167, 284, 484, 835, 1433, 2467, 4250, 7345, 12700, 22004, 38154, 66266, 115224, 200623, 349654, 610126, 1065739, 1863547, 3261672, 5714277, 10020092, 17586014, 30890654, 54305289, 95542387, 168221056, 296401979
Offset: 4

Views

Author

Joerg Arndt and Alois P. Heinz, Jun 29 2014

Keywords

Examples

			a(7) = 1:
      o
     /|\
    o o o
   /|\
  o o o

Links

Crossrefs

Column k=3 of A244454.
Cf. A244532.

Programs

  • Maple
    b:= proc(n, i, t, k) option remember; `if`(n=0, `if`(t in [0, k],
      1, 0), `if`(i<1 or t>n, 0, add(binomial(b((i-1)$2, k$2)+j-1, j)*
      b(n-i*j, i-1, max(0,t-j), k), j=0..n/i)))
    end:
a:= n-> b(n-1$2, 3$2) -b(n-1$2, 4$2):
seq(a(n), n=4..45);
  • Mathematica
    b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, If[t == 0 || t == k, 1, 0], If[i < 1, 0, Sum[Binomial[b[i - 1, i - 1, k, k] + j - 1, j]*b[n - i*j, i - 1, Max[0, t - j], k], {j, 0, n/i}]] // FullSimplify]; a[n_] := b[n - 1, n - 1, 3, 3] - b[n - 1, n - 1, 4, 4]; Table[a[n], {n, 4, 45}] (* Jean-François Alcover, Feb 06 2015, after Maple *)

Formula

a(n) ~ c * d^n / n^(3/2), where d = 1.8239199077079..., c = 0.49573400799... . - Vaclav Kotesovec, Jul 11 2014

A244458 Number of unlabeled rooted trees with n nodes such that the minimal outdegree of inner nodes equals 4.

Original entry on oeis.org

1, 0, 0, 0, 1, 2, 2, 2, 4, 7, 12, 16, 25, 38, 61, 94, 147, 227, 356, 550, 862, 1345, 2113, 3299, 5168, 8091, 12721, 19981, 31421, 49384, 77761, 122487, 193151, 304623, 480852, 759367, 1200150, 1897594, 3002329, 4752436, 7527155, 11927290, 18909719, 29993579
Offset: 5

Views

Author

Joerg Arndt and Alois P. Heinz, Jun 29 2014

Keywords

Examples

			a(9) = 1:
         o
      / ( ) \
     o  o o  o
   /( )\
  o o o o

Links

Crossrefs

Column k=4 of A244454.
Cf. A244533.

Programs

  • Maple
    b:= proc(n, i, t, k) option remember; `if`(n=0, `if`(t in [0, k],
      1, 0), `if`(i<1 or t>n, 0, add(binomial(b((i-1)$2, k$2)+j-1, j)*
      b(n-i*j, i-1, max(0,t-j), k), j=0..n/i)))
    end:
a:= n-> b(n-1$2, 4$2) -b(n-1$2, 5$2):
seq(a(n), n=5..50);
  • Mathematica
    b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, If[t == 0 || t == k, 1, 0], If[i < 1, 0, Sum[Binomial[b[i - 1, i - 1, k, k] + j - 1, j]*b[n - i*j, i - 1, Max[0, t - j], k], {j, 0, n/i}]] // FullSimplify]; a[n_] := b[n - 1, n - 1, 4, 4] - b[n - 1, n - 1, 5, 5]; Table[a[n], {n, 5, 50}] (* Jean-François Alcover, Feb 06 2015, after Maple *)

A244459 Number of unlabeled rooted trees with n nodes such that the minimal outdegree of inner nodes equals 5.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 2, 2, 2, 2, 4, 7, 12, 16, 21, 29, 43, 65, 99, 142, 206, 297, 436, 641, 945, 1383, 2029, 2976, 4378, 6432, 9464, 13913, 20495, 30205, 44547, 65670, 96846, 142857, 210941, 311636, 460613, 680848, 1006682, 1488915, 2203324, 3261840, 4830671
Offset: 6

Views

Author

Joerg Arndt and Alois P. Heinz, Jun 29 2014

Keywords

Links

Crossrefs

Column k=5 of A244454.
Cf. A244534.

Programs

  • Maple
    b:= proc(n, i, t, k) option remember; `if`(n=0, `if`(t in [0, k],
      1, 0), `if`(i<1 or t>n, 0, add(binomial(b((i-1)$2, k$2)+j-1, j)*
      b(n-i*j, i-1, max(0,t-j), k), j=0..n/i)))
    end:
a:= n-> b(n-1$2, 5$2) -b(n-1$2, 6$2):
seq(a(n), n=6..55);
  • Mathematica
    b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, If[t == 0 || t == k, 1, 0], If[i < 1, 0, Sum[Binomial[b[i - 1, i - 1, k, k] + j - 1, j]*b[n - i*j, i - 1, Max[0, t - j], k], {j, 0, n/i}]] // FullSimplify]; a[n_] := b[n - 1, n - 1, 5, 5] - b[n - 1, n - 1, 6, 6]; Table[a[n], {n, 6, 55}] (* Jean-François Alcover, Feb 06 2015, after Maple *)

A244460 Number of unlabeled rooted trees with n nodes such that the minimal outdegree of inner nodes equals 6.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 2, 2, 2, 2, 2, 4, 7, 12, 16, 21, 25, 34, 47, 70, 103, 147, 201, 276, 377, 527, 743, 1057, 1486, 2088, 2911, 4073, 5704, 8027, 11290, 15897, 22340, 31411, 44159, 62165, 87516, 123296, 173642, 244636, 344684, 485976, 685362, 966971, 1364301
Offset: 7

Views

Author

Joerg Arndt and Alois P. Heinz, Jun 29 2014

Keywords

Links

Crossrefs

Column k=6 of A244454.
Cf. A244535.

Programs

  • Maple
    b:= proc(n, i, t, k) option remember; `if`(n=0, `if`(t in [0, k],
      1, 0), `if`(i<1 or t>n, 0, add(binomial(b((i-1)$2, k$2)+j-1, j)*
      b(n-i*j, i-1, max(0,t-j), k), j=0..n/i)))
    end:
a:= n-> b(n-1$2, 6$2) -b(n-1$2, 7$2):
seq(a(n), n=7..60);
  • Mathematica
    b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, If[t == 0 || t == k, 1, 0], If[i < 1, 0, Sum[Binomial[b[i - 1, i - 1, k, k] + j - 1, j]*b[n - i*j, i - 1, Max[0, t - j], k], {j, 0, n/i}]] // FullSimplify]; a[n_] := b[n - 1, n - 1, 6, 6] - b[n - 1, n - 1, 7, 7]; Table[a[n], {n, 7, 60}] (* Jean-François Alcover, Feb 06 2015, after Maple *)

A244461 Number of unlabeled rooted trees with n nodes such that the minimal outdegree of inner nodes equals 7.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 1, 2, 2, 2, 2, 2, 2, 4, 7, 12, 16, 21, 25, 30, 38, 52, 74, 108, 151, 206, 271, 356, 468, 629, 855, 1180, 1620, 2212, 2991, 4030, 5420, 7320, 9922, 13508, 18396, 25049, 34032, 46194, 62653, 85051, 115548, 157168, 213852, 291046, 395990
Offset: 8

Views

Author

Joerg Arndt and Alois P. Heinz, Jun 29 2014

Keywords

Links

Crossrefs

Column k=7 of A244454.
Cf. A244536.

Programs

  • Maple
    b:= proc(n, i, t, k) option remember; `if`(n=0, `if`(t in [0, k],
      1, 0), `if`(i<1 or t>n, 0, add(binomial(b((i-1)$2, k$2)+j-1, j)*
      b(n-i*j, i-1, max(0,t-j), k), j=0..n/i)))
    end:
a:= n-> b(n-1$2, 7$2) -b(n-1$2, 8$2):
seq(a(n), n=8..65);
  • Mathematica
    b[n_, i_, t_, k_] := b[n, i, t, k] = If[n == 0, If[t == 0 || t == k, 1, 0], If[i < 1, 0, Sum[Binomial[b[i - 1, i - 1, k, k] + j - 1, j]*b[n - i*j, i - 1, Max[0, t - j], k], {j, 0, n/i}]] // FullSimplify]; a[n_] := b[n - 1, n - 1, 7, 7] - b[n - 1, n - 1, 8, 8]; Table[a[n], {n, 8, 65}] (* Jean-François Alcover, Feb 06 2015, after Maple *)

A244462 Number of unlabeled rooted trees with n nodes such that the minimal outdegree of inner nodes equals 8.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 1, 2, 2, 2, 2, 2, 2, 2, 4, 7, 12, 16, 21, 25, 30, 34, 43, 56, 79, 112, 156, 210, 276, 351, 447, 570, 741, 978, 1314, 1764, 2367, 3137, 4131, 5398, 7058, 9236, 12161, 16077, 21340, 28326, 37573, 49696, 65623, 86526, 114122, 150624
Offset: 9

Views

Author

Joerg Arndt and Alois P. Heinz, Jun 29 2014

Keywords

Links

Crossrefs

Column k=8 of A244454.
Cf. A244537.

Programs

  • Maple
    b:= proc(n, i, t, k) option remember; `if`(n=0, `if`(t in [0, k],
      1, 0), `if`(i<1 or t>n, 0, add(binomial(b((i-1)$2, k$2)+j-1, j)*
      b(n-i*j, i-1, max(0,t-j), k), j=0..n/i)))
    end:
a:= n-> b(n-1$2, 8$2) -b(n-1$2, 9$2):
seq(a(n), n=9..70);
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