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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.
7
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
0,9
COMMENTS
Homeomorphically irreducible trees are trees without vertices of degree 2. All non-leaf nodes then have degree >= 3.
LINKS
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.
EXAMPLE
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;
...
PROG
(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])); }
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.
Sequence in context: A029584 A318931 A266878 * A199469 A266832 A373570
KEYWORD
nonn,tabl
AUTHOR
Andrew Howroyd, Dec 16 2020
STATUS
approved