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.

%I A339780 #14 Apr 20 2023 14:55:38
%S A339780 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,
%T A339780 0,13,354,2673,8512,13100,9672,2752,0,32,1491,17571,84312,199820,
%U A339780 248904,156492,39208,0,73,6504,117365,814184,2782970,5194580,5408620,2953792,660032
%N A339780 Triangle read by rows: T(n,k) is the number of homeomorphically irreducible leaf colored trees with n leaves using exactly k colors.
%C A339780 Homeomorphically irreducible trees are trees without vertices of degree 2. All non-leaf nodes then have degree >= 3.
%H A339780 Andrew Howroyd, <a href="/A339780/b339780.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%H A339780 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.
%e A339780 Triangle begins:
%e A339780   1;
%e A339780   0,  1;
%e A339780   0,  1,    1;
%e A339780   0,  1,    2,     1;
%e A339780   0,  2,    7,     9,     4;
%e A339780   0,  3,   24,    63,    68,     26;
%e A339780   0,  7,   91,   412,   812,    720,    236;
%e A339780   0, 13,  354,  2673,  8512,  13100,   9672,   2752;
%e A339780   0, 32, 1491, 17571, 84312, 199820, 248904, 156492, 39208;
%e A339780   ...
%o A339780 (PARI) \\ here U(n,k) is A339779 as vector.
%o A339780 EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
%o A339780 R(n, k)={my(v=[k]); for(n=2, n, v=concat(v, EulerT(concat(v, [0]))[n])); v}
%o A339780 U(n, k)={my(g=x*Ser(R(n,k))); Vec(1 + g + k*x*g - g^2)}
%o A339780 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 A339780 { my(T=M(8)); for(n=1, #T~, print(T[n,1..n])); }
%Y A339780 Columns k=1..4 are A007827(n>0), A339785, A339786, A339787.
%Y A339780 Main diagonal is A000311(n>0).
%Y A339780 Row sums are A339781.
%Y A339780 Cf. A319376 (planted), A339650 (degree <= 3), A339779.
%K A339780 nonn,tabl
%O A339780 0,9
%A A339780 _Andrew Howroyd_, Dec 16 2020