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.

A318754 Number T(n,k) of rooted trees with n nodes such that k equals the maximal number of subtrees extending from the same node and having the same number of nodes; 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, 2, 1, 1, 0, 3, 4, 1, 1, 0, 6, 9, 3, 1, 1, 0, 12, 22, 9, 3, 1, 1, 0, 25, 54, 23, 8, 3, 1, 1, 0, 51, 139, 60, 23, 8, 3, 1, 1, 0, 111, 346, 166, 61, 22, 8, 3, 1, 1, 0, 240, 892, 447, 167, 61, 22, 8, 3, 1, 1, 0, 533, 2290, 1219, 461, 168, 60, 22, 8, 3, 1, 1
Offset: 1

Views

Author

Alois P. Heinz, Sep 02 2018

Keywords

Comments

T(n,k) is defined for n,k >= 0. The triangle contains only the terms with k < n. T(n,k) = 0 for k >= n.

Examples

			Triangle T(n,k) begins:
  1;
  0,   1;
  0,   1,   1;
  0,   2,   1,   1;
  0,   3,   4,   1,  1;
  0,   6,   9,   3,  1,  1;
  0,  12,  22,   9,  3,  1, 1;
  0,  25,  54,  23,  8,  3, 1, 1;
  0,  51, 139,  60, 23,  8, 3, 1, 1;
  0, 111, 346, 166, 61, 22, 8, 3, 1, 1;

Links

Crossrefs

Columns k=0-10 give: A063524, A032305 (for n>1), A318817, A318818, A318819, A318820, A318821, A318822, A318823, A318824, A318825.
Row sums give A000081.
T(2n+2,n+1) give A255705.
Cf. A318753.

Programs

  • Maple
    g:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      binomial(g(i-1$2, k)+j-1, j)*g(n-i*j, i-1, k), j=0..min(k, n/i))))
    end:
T:= (n, k)-> g(n-1$2, k) -`if`(k=0, 0, g(n-1$2, k-1)):
seq(seq(T(n, k), k=0..n-1), n=1..14);
  • Mathematica
    g[n_, i_, k_] := g[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[g[i - 1, i - 1, k] + j - 1, j]*g[n - i*j, i - 1, k], {j, 0, Min[k, n/i]}]]];
T[n_, k_] := g[n - 1, n - 1, k] - If[k == 0, 0, g[n - 1, n - 1, k - 1]];
Table[T[n, k], {n, 1, 14}, {k, 0, n - 1}] // Flatten (* Jean-François Alcover, May 27 2019, after Alois P. Heinz *)

Formula

T(n,k) = A318753(n,k) - A318753(n,k-1) for k > 0, A(n,0) = A063524(n).

A318758 Number T(n,k) of rooted trees with n nodes such that k equals the maximal number of isomorphic subtrees extending from the same node; 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, 2, 1, 1, 0, 3, 4, 1, 1, 0, 6, 9, 3, 1, 1, 0, 12, 22, 9, 3, 1, 1, 0, 25, 54, 23, 8, 3, 1, 1, 0, 52, 138, 60, 23, 8, 3, 1, 1, 0, 113, 346, 164, 61, 22, 8, 3, 1, 1, 0, 247, 889, 443, 167, 61, 22, 8, 3, 1, 1, 0, 548, 2285, 1209, 461, 168, 60, 22, 8, 3, 1, 1
Offset: 1

Views

Author

Alois P. Heinz, Sep 02 2018

Keywords

Comments

T(n,k) is defined for n,k >= 0. The triangle contains only the terms with k < n. T(n,k) = 0 for k >= n.

Examples

			Triangle T(n,k) begins:
  1;
  0,   1;
  0,   1,   1;
  0,   2,   1,   1;
  0,   3,   4,   1,  1;
  0,   6,   9,   3,  1,  1;
  0,  12,  22,   9,  3,  1, 1;
  0,  25,  54,  23,  8,  3, 1, 1;
  0,  52, 138,  60, 23,  8, 3, 1, 1;
  0, 113, 346, 164, 61, 22, 8, 3, 1, 1;

Links

Crossrefs

Columns k=0-10 give: A063524, A004111 (for n>1), A318859, A318860, A318861, A318862, A318863, A318864, A318865, A318866, A318867.
Row sums give A000081.
T(2n+2,n+1) gives A255705.
Cf. A318754, A318757.

Programs

  • Maple
    h:= proc(n, m, t, k) option remember; `if`(m=0, binomial(n+t, t),
      `if`(n=0, 0, add(h(n-1, m-j, t+1, k), j=1..min(k, m))))
    end:
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1, k)*h(A(i, k), j, 0, k), j=0..n/i)))
    end:
A:= (n, k)-> `if`(n<2, n, b(n-1$2, k)):
T:= (n, k)-> A(n, k)-`if`(k=0, 0, A(n, k-1)):
seq(seq(T(n, k), k=0..n-1), n=1..14);
  • Mathematica
    h[n_, m_, t_, k_] := h[n, m, t, k] = If[m == 0, Binomial[n + t, t], If[n == 0, 0, Sum[h[n - 1, m - j, t + 1, k], {j, 1, Min[k, m]}]]];
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1, k]*h[A[i, k], j, 0, k], {j, 0, n/i}]]];
A[n_, k_] := If[n < 2, n, b[n - 1, n - 1, k]];
T[n_, k_] := A[n, k] - If[k == 0, 0, A[n, k - 1]];
Table[T[n, k], {n, 1, 14}, {k, 0, n - 1}] // Flatten (* Jean-François Alcover, May 11 2019, after Alois P. Heinz *)

Formula

T(n,k) = A318757(n,k) - A318757(n,k-1) for k > 0, A(n,0) = A063524(n).

A255704 Number T(n,k) of n-node rooted trees in which the maximal number of nodes in paths starting at a leaf and ending at the first branching node or at the root equals k; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 4, 3, 1, 1, 0, 8, 7, 3, 1, 1, 0, 17, 18, 8, 3, 1, 1, 0, 36, 45, 21, 8, 3, 1, 1, 0, 79, 116, 56, 22, 8, 3, 1, 1, 0, 175, 298, 152, 59, 22, 8, 3, 1, 1, 0, 395, 776, 413, 163, 60, 22, 8, 3, 1, 1, 0, 899, 2025, 1131, 450, 166, 60, 22, 8, 3, 1, 1
Offset: 1

Views

Author

Alois P. Heinz, Mar 02 2015

Keywords

Examples

			:    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
:                                           |
: T(6,3) = 7                                o
Triangle T(n,k) begins:
  1;
  0,   1;
  0,   1,   1;
  0,   2,   1,   1;
  0,   4,   3,   1,  1;
  0,   8,   7,   3,  1,  1;
  0,  17,  18,   8,  3,  1, 1;
  0,  36,  45,  21,  8,  3, 1, 1;
  0,  79, 116,  56, 22,  8, 3, 1, 1;
  0, 175, 298, 152, 59, 22, 8, 3, 1, 1;

Links

Crossrefs

Columns k=1-10 give: A063524, A002955 (for n>1), A318899, A318900, A318901, A318902, A318903, A318904, A318905, A318906.
Row sums give A000081.
T(2*n+1,n+1) gives A255705.
Cf. A255636.

Programs

  • Maple
    with(numtheory):
g:= proc(n, k) option remember; `if`(n=0, 1, add(add(d*(g(d-1, k)-
      `if`(d=k, 1, 0)), d=divisors(j))*g(n-j, k), j=1..n)/n)
    end:
T:= (n, k)-> g(n-1, k) -`if`(k=1, 0, g(n-1, k-1)):
seq(seq(T(n, k), k=1..n), n=1..14);
  • Mathematica
    g[n_, k_] := g[n, k] = If[n == 0, 1, Sum[DivisorSum[j, #*(g[#-1, k] - If[# == k, 1, 0])&] * g[n-j, k], {j, 1, n}]/n];
T[n_, k_] := g[n-1, k] - If[k == 1, 0, g[n-1, k-1]];
Table[T[n, k], {n, 1, 14}, {k, 1, n}] // Flatten (* Jean-François Alcover, Mar 24 2017, translated from Maple *)

Formula

T(n,1) = A255636(n,1), T(n,k) = A255636(n,k) - A255636(n,k-1) for k>1.
Showing 1-3 of 3 results.

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