A339780 - OEIS

%I #14 Apr 20 2023 14:55:38

%S 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 0,13,354,2673,8512,13100,9672,2752,0,32,1491,17571,84312,199820,

%U 248904,156492,39208,0,73,6504,117365,814184,2782970,5194580,5408620,2953792,660032

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

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

%H Andrew Howroyd, <a href="/A339780/b339780.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)

%H 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 Triangle begins:

%e   1;

%e   0,  1;

%e   0,  1,    1;

%e   0,  1,    2,     1;

%e   0,  2,    7,     9,     4;

%e   0,  3,   24,    63,    68,     26;

%e   0,  7,   91,   412,   812,    720,    236;

%e   0, 13,  354,  2673,  8512,  13100,   9672,   2752;

%e   0, 32, 1491, 17571, 84312, 199820, 248904, 156492, 39208;

%e   ...

%o (PARI) \\ here U(n,k) is A339779 as vector.

%o EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}

%o R(n, k)={my(v=[k]); for(n=2, n, v=concat(v, EulerT(concat(v, [0]))[n])); v}

%o U(n, k)={my(g=x*Ser(R(n,k))); Vec(1 + g + k*x*g - g^2)}

%o 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 { my(T=M(8)); for(n=1, #T~, print(T[n,1..n])); }

%Y Columns k=1..4 are A007827(n>0), A339785, A339786, A339787.

%Y Main diagonal is A000311(n>0).

%Y Row sums are A339781.

%Y Cf. A319376 (planted), A339650 (degree <= 3), A339779.

%K nonn,tabl

%O 0,9

%A _Andrew Howroyd_, Dec 16 2020