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A339779 Array read by antidiagonals: T(n,k) is the number of homeomorphically irreducible leaf colored trees with n leaves of k colors.

%I A339779 #20 Apr 20 2023 14:56:30
%S A339779 1,1,0,1,1,0,1,2,1,0,1,3,3,1,0,1,4,6,4,2,0,1,5,10,10,11,3,0,1,6,15,20,
%T A339779 36,30,7,0,1,7,21,35,90,144,105,13,0,1,8,28,56,190,476,706,380,32,0,1,
%U A339779 9,36,84,357,1251,3034,3774,1555,73,0,1,10,45,120,616,2814,9845,21380,22140,6650,190,0
%N A339779 Array read by antidiagonals: T(n,k) is the number of homeomorphically irreducible leaf colored trees with n leaves of k colors.
%C A339779 Homeomorphically irreducible trees are trees without vertices of degree 2. All non-leaf nodes then have degree >= 3.
%C A339779 Not all colors need to be used.
%C A339779 The Johnson reference has a mistake in formula 4.3. In particular, the final term should be subtracted rather than added. Compare with the first formula given here. The table of results given in the reference is consequently also incorrect.
%H A339779 Andrew Howroyd, <a href="/A339779/b339779.txt">Table of n, a(n) for n = 0..1325</a>
%H A339779 Virginia Perkins Johnson, <a href="https://people.math.sc.edu/czabarka/Theses/JohnsonThesis.pdf">Enumeration Results on Leaf Labeled Trees</a>, Ph. D. Dissertation, Univ. South Carolina, 2012. [Table 4.3 is an incorrect version of this table].
%F A339779 T(n,k) = k*g(n-1,k) + g(n,k) - Sum_{j=1..n-1} g(j,k)*g(n-j,k) for n > 1 where g(n,k) is A319254(n,k).
%F A339779 G.f. of k-th column: 1 + k*x*r(x) + r(x) - r(x)^2 where r(x) is the g.f. of the k-th column of A319254.
%e A339779 Array begins:
%e A339779 ============================================================
%e A339779 n\k| 0  1    2      3       4       5        6         7
%e A339779 ---+--------------------------------------------------------
%e A339779 0  | 1  1    1      1       1       1        1         1 ...
%e A339779 1  | 0  1    2      3       4       5        6         7 ...
%e A339779 2  | 0  1    3      6      10      15       21        28 ...
%e A339779 3  | 0  1    4     10      20      35       56        84 ...
%e A339779 4  | 0  2   11     36      90     190      357       616 ...
%e A339779 5  | 0  3   30    144     476    1251     2814      5656 ...
%e A339779 6  | 0  7  105    706    3034    9845    26383     61572 ...
%e A339779 7  | 0 13  380   3774   21380   85995   274800    744556 ...
%e A339779 8  | 0 32 1555  22140  163670  812160  3086481   9692480 ...
%e A339779 9  | 0 73 6650 137096 1322960 8092945 36550458 132954360 ...
%e A339779      ...
%o A339779 (PARI) \\ here R(n,k) is k-th column of A319254.
%o A339779 EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
%o A339779 R(n, k)={my(v=[k]); for(n=2, n, v=concat(v, EulerT(concat(v, [0]))[n])); v}
%o A339779 U(n, k)={my(g=x*Ser(R(n,k))); Vec(1 + g + k*x*g - g^2)}
%o A339779 {my(T=Mat(vector(9, k, U(8, k-1)~))); for(n=1, #T~, print(T[n, ]))}
%Y A339779 Columns k=1..4 are A007827, A339782, A339783, A339784.
%Y A339779 Cf. A319254 (planted), A339649 (degree <= 3), A339780.
%K A339779 nonn,tabl
%O A339779 0,8
%A A339779 _Andrew Howroyd_, Dec 16 2020