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.

A339780 Triangle read by rows: T(n,k) is the number of homeomorphically irreducible leaf colored trees with n leaves using exactly k colors.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 2, 7, 9, 4, 0, 3, 24, 63, 68, 26, 0, 7, 91, 412, 812, 720, 236, 0, 13, 354, 2673, 8512, 13100, 9672, 2752, 0, 32, 1491, 17571, 84312, 199820, 248904, 156492, 39208, 0, 73, 6504, 117365, 814184, 2782970, 5194580, 5408620, 2953792, 660032
Offset: 0

Views

Author

Andrew Howroyd, Dec 16 2020

Keywords

Comments

Homeomorphically irreducible trees are trees without vertices of degree 2. All non-leaf nodes then have degree >= 3.

Examples

			Triangle begins:
  1;
  0,  1;
  0,  1,    1;
  0,  1,    2,     1;
  0,  2,    7,     9,     4;
  0,  3,   24,    63,    68,     26;
  0,  7,   91,   412,   812,    720,    236;
  0, 13,  354,  2673,  8512,  13100,   9672,   2752;
  0, 32, 1491, 17571, 84312, 199820, 248904, 156492, 39208;
  ...

Links

Crossrefs

Columns k=1..4 are A007827(n>0), A339785, A339786, A339787.
Main diagonal is A000311(n>0).
Row sums are A339781.
Cf. A319376 (planted), A339650 (degree <= 3), A339779.

Programs

  • PARI
    \\ here U(n,k) is A339779 as vector.
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
R(n, k)={my(v=[k]); for(n=2, n, v=concat(v, EulerT(concat(v, [0]))[n])); v}
U(n, k)={my(g=x*Ser(R(n,k))); Vec(1 + g + k*x*g - g^2)}
M(n, m=n)={my(v=vector(m+1, k, U(n, k-1)~)); Mat(vector(m+1, k, k--; sum(i=0, k, (-1)^(k-i)*binomial(k, i)*v[1+i])))}
{ my(T=M(8)); for(n=1, #T~, print(T[n,1..n])); }

A339649 Array read by antidiagonals: T(n,k) is the number of leaf colored trees with n leaves of k colors and all non-leaf nodes having degree 3.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 1, 0, 1, 4, 6, 4, 1, 0, 1, 5, 10, 10, 6, 1, 0, 1, 6, 15, 20, 21, 12, 2, 0, 1, 7, 21, 35, 55, 63, 31, 2, 0, 1, 8, 28, 56, 120, 220, 227, 78, 4, 0, 1, 9, 36, 84, 231, 600, 1040, 891, 234, 6, 0, 1, 10, 45, 120, 406, 1386, 3530, 5480, 3876, 722, 11, 0
Offset: 0

Views

Author

Andrew Howroyd, Dec 14 2020

Keywords

Comments

Not all k colors need to be used. The total number of nodes will be 2n-1.
See table 4.1 in the Johnson reference.

Examples

			Array begins:
======================================================
n\k| 0 1   2     3      4       5       6        7
---+--------------------------------------------------
0  | 1 1   1     1      1       1       1        1 ...
1  | 0 1   2     3      4       5       6        7 ...
2  | 0 1   3     6     10      15      21       28 ...
3  | 0 1   4    10     20      35      56       84 ...
4  | 0 1   6    21     55     120     231      406 ...
5  | 0 1  12    63    220     600    1386     2842 ...
6  | 0 2  31   227   1040    3530    9772    23366 ...
7  | 0 2  78   891   5480   23250   77112   214718 ...
8  | 0 4 234  3876  31420  165510  655599  2122099 ...
9  | 0 6 722 17790 190360 1243825 5878446 22102577 ...
     ...

Links

Crossrefs

Columns k=1..4 are A129860, A220826, A220827, A220828.
Cf. A319539 (rooted), A339650, A339779.

Programs

  • PARI
    \\ here U(n,k) gives column k as a vector.
R(n, k)={my(v=vector(n)); v[1]=k; for(n=2, n, v[n]=sum(j=1, (n-1)\2, v[j]*v[n-j]) + if(n%2, 0, binomial(v[n/2]+1, 2))); v}
U(n, k)={my(g=x*Ser(R(n,k))); Vec(1 + g + (subst(g + O(x*x^(n\3)), x, x^3) - g^3)/3)}
{my(T=Mat(vector(8, k, U(8, k-1)~))); for(n=1, #T~, print(T[n,]))}

Formula

G.f. of column k: 1 + R(x) + (R(x^3) - R(x)^3)/3 where R(x) is the g.f. of column k of A319539.

A339782 Number of homeomorphically irreducible leaf colored trees with n leaves of 2 colors.

Original entry on oeis.org

1, 2, 3, 4, 11, 30, 105, 380, 1555, 6650, 30091, 140592, 677396, 3336280, 16752485, 85444832, 441757114, 2310561412, 12208670550, 65088056150, 349773085560, 1893030193232, 10311145350621, 56489805467868, 311113902447810, 1721697849252838, 9569947189448379, 53410284955104654
Offset: 0

Views

Author

Andrew Howroyd, Dec 16 2020

Keywords

Examples

			There are two trees for a(4):
          o         o    o
          |         |    |
      o---o---o     o----o
          |         |    |
          o         o    o
The first of these can have the 4 leaves colored in 5 indistinguishable ways and the second can have the 4 leaves colored in 6 indistinguishable ways so a(4) = 5 + 6 = 11.

Links

Crossrefs

Column k=2 of A339779.

Programs

  • PARI
    U(25, 2) \\ See A339779 for U(n,k).

A339783 Number of homeomorphically irreducible leaf colored trees with n leaves of 3 colors.

Original entry on oeis.org

1, 3, 6, 10, 36, 144, 706, 3774, 22140, 137096, 888522, 5950434, 40936518, 287748366, 2059386024, 14964163584, 110160504906, 820172543046, 6167144620494, 46780206203250, 357620668556700, 2753050958956812, 21327412744225182, 166164346278018822, 1301341878493196986
Offset: 0

Views

Author

Andrew Howroyd, Dec 16 2020

Keywords

Links

Crossrefs

Column k=3 of A339779.

Programs

  • PARI
    U(25, 3) \\ See A339779 for U(n,k).

A339784 Number of homeomorphically irreducible leaf colored trees with n leaves of 4 colors.

Original entry on oeis.org

1, 4, 10, 20, 90, 476, 3034, 21380, 163670, 1322960, 11156014, 97142260, 868031736, 7921646244, 73577937258, 693691932892, 6624644271386, 63974219412076, 623878427017208, 6137012579638796, 60836810198042200, 607270688878149596, 6099719973597772370
Offset: 0

Views

Author

Andrew Howroyd, Dec 16 2020

Keywords

Links

Crossrefs

Column k=4 of A339779.

Programs

  • PARI
    U(25, 4) \\ See A339779 for U(n,k).
Showing 1-5 of 5 results.

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