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A339650 Triangle read by rows: T(n,k) is the number of trees with n leaves of exactly k colors and all non-leaf nodes having degree 3.

%I A339650 #18 Apr 20 2023 14:56:34
%S A339650 1,0,1,0,1,1,0,1,2,1,0,1,4,6,3,0,1,10,30,36,15,0,2,27,140,310,300,105,
%T A339650 0,2,74,663,2376,3990,3150,945,0,4,226,3186,17304,44850,59805,39690,
%U A339650 10395,0,6,710,15642,123508,462735,925890,1018710,582120,135135
%N A339650 Triangle read by rows: T(n,k) is the number of trees with n leaves of exactly k colors and all non-leaf nodes having degree 3.
%C A339650 See table 4.2 in the Johnson reference.
%H A339650 Andrew Howroyd, <a href="/A339650/b339650.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%H A339650 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.
%F A339650 T(n,k) = Sum_{i=0..k} (-1)^(k-i)*binomial(k,i)*A339649(n,i).
%e A339650 Triangle begins:
%e A339650   1;
%e A339650   0, 1;
%e A339650   0, 1,   1;
%e A339650   0, 1,   2,    1;
%e A339650   0, 1,   4,    6,     3;
%e A339650   0, 1,  10,   30,    36,    15;
%e A339650   0, 2,  27,  140,   310,   300,   105;
%e A339650   0, 2,  74,  663,  2376,  3990,  3150,   945;
%e A339650   0, 4, 226, 3186, 17304, 44850, 59805, 39690, 10395;
%e A339650   ...
%o A339650 (PARI) \\ here U(n,k) is column k of A339649 as a vector.
%o A339650 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}
%o A339650 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)}
%o A339650 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])))}
%o A339650 {my(T=M(10)); for(n=1, #T~, print(T[n, ][1..n]))}
%Y A339650 Columns k=1..4 are A129860, A220829, A220830, A220831.
%Y A339650 Main diagonal is A001147(n-2) for n >= 2.
%Y A339650 Row sums are A339651.
%Y A339650 Cf. A319541 (rooted), A339649, A339780.
%K A339650 nonn,tabl
%O A339650 0,9
%A A339650 _Andrew Howroyd_, Dec 14 2020