A330763 Triangle read by rows: T(n,k) is the number of series-reduced rooted trees whose leaves are sets of colors with a total of n elements using exactly k colors.

1, 1, 2, 2, 8, 8, 5, 41, 90, 58, 12, 204, 852, 1264, 612, 33, 1046, 7428, 19568, 21510, 8374, 90, 5456, 62682, 262912, 496270, 431040, 140408, 261, 29165, 523167, 3291021, 9520220, 13884960, 9947294, 2785906, 766, 158792, 4358182, 39636784, 165204730, 360421716, 426677440, 259854304, 63830764

Offset: 1

Author

Andrew Howroyd, Dec 29 2019

Keywords

• nonn
• tabl

Examples

			Triangle begins:
    1;
    1,     2;
    2,     8,      8;
    5,    41,     90,      58;
   12,   204,    852,    1264,     612;
   33,  1046,   7428,   19568,   21510,     8374;
   90,  5456,  62682,  262912,  496270,   431040,  140408;
  261, 29165, 523167, 3291021, 9520220, 13884960, 9947294, 2785906;
  ...
The T(3,2) = 8 trees are: ((1)(12)), ((2)(12)), ((1)(2)(2)), ((1)(1)(2)), ((1)((2)(2))), ((1)((1)(2))), ((2)((1)(2))), ((2)((1)(1))).
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)

Crossrefs

Column 1 is A000669.

Main diagonal is A005804.

Row sums are A330764.

Cf. A330762 (leaves are multisets).

Programs

PARI

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
R(n, k)={my(v=[]); for(n=1, n, v=concat(v, EulerT(concat(v, [binomial(k,n)]))[n])); v}
M(n)={my(v=vector(n, k, R(n, k)~)); Mat(vector(n, k, sum(i=1, k, (-1)^(k-i)*binomial(k, i)*v[i])))}
{my(T=M(10)); for(n=1, #T~, print(T[n, 1..n]))} \\ Andrew Howroyd, Dec 29 2019