A339785 -id:A339785 - cp's OEIS frontend

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

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

Andrew Howroyd, Dec 16 2020

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

			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;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
Virginia Perkins Johnson, Enumeration Results on Leaf Labeled Trees, Ph. D. Dissertation, Univ. South Carolina, 2012.

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.

\\ 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])); }

A339786 Number of homeomorphically irreducible leaf colored trees with n leaves using exactly 3 colors.

Original entry on oeis.org

0, 0, 1, 9, 63, 412, 2673, 17571, 117365, 798819, 5530122, 38908380, 277750749, 2009160864, 14707923021, 108835512411, 813241695330, 6130521151377, 46584949832013, 356571373433217, 2747371943624943, 21296479544449677, 165994877608025730, 1300408539157086640

Offset: 1

Andrew Howroyd, Dec 18 2020

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..500

Column k=3 of A339780.

Cf. A007827, A339782, A339783, A339785.

my(N=25); (U(N,3) - 3*U(N,2) + 3*U(N,1))[2..1+N] \\ See A339780 for U(n, k).

a(n) = A339783(n) - 3*A339782(n) + 3*A007827(n).

A339787 Number of homeomorphically irreducible leaf colored trees with n leaves using exactly 4 colors.

Original entry on oeis.org

0, 0, 0, 4, 68, 812, 8512, 84312, 814184, 7781712, 74182124, 708344640, 6790655496, 65440865012, 634347822304, 6186652422650, 60707391493004, 599283097168488, 5950282272766412, 59408426130151164, 596269843123151304, 6014472189177940224, 60952019560703982452

Offset: 1

Andrew Howroyd, Dec 18 2020

nonn

			There are 2 homeomorphically reduced trees with 4 leafs:
            o          o   o
            |          |   |
        o---o---o      o---o
            |          |   |
            o          o   o
The leaves of the first tree can be colored in 1 way using all four colors and the second can be colored in 3 ways, so a(4) = 1 + 3 = 4.

Andrew Howroyd, Table of n, a(n) for n = 1..500

Column k=4 of A339780.

Cf. A339785, A339786.

my(N=25); M(N,4)[2..1+N, 5]~ \\ See A339780 for M(n, m).

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

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A339786 Number of homeomorphically irreducible leaf colored trees with n leaves using exactly 3 colors.

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A339787 Number of homeomorphically irreducible leaf colored trees with n leaves using exactly 4 colors.

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