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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.

%I A330763 #8 Jan 09 2020 19:43:53
%S A330763 1,1,2,2,8,8,5,41,90,58,12,204,852,1264,612,33,1046,7428,19568,21510,
%T A330763 8374,90,5456,62682,262912,496270,431040,140408,261,29165,523167,
%U A330763 3291021,9520220,13884960,9947294,2785906,766,158792,4358182,39636784,165204730,360421716,426677440,259854304,63830764
%N 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.
%H A330763 Andrew Howroyd, <a href="/A330763/b330763.txt">Table of n, a(n) for n = 1..1275</a> (first 50 rows)
%e A330763 Triangle begins:
%e A330763     1;
%e A330763     1,     2;
%e A330763     2,     8,      8;
%e A330763     5,    41,     90,      58;
%e A330763    12,   204,    852,    1264,     612;
%e A330763    33,  1046,   7428,   19568,   21510,     8374;
%e A330763    90,  5456,  62682,  262912,  496270,   431040,  140408;
%e A330763   261, 29165, 523167, 3291021, 9520220, 13884960, 9947294, 2785906;
%e A330763   ...
%e A330763 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))).
%o A330763 (PARI)
%o A330763 EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
%o A330763 R(n, k)={my(v=[]); for(n=1, n, v=concat(v, EulerT(concat(v, [binomial(k,n)]))[n])); v}
%o A330763 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])))}
%o A330763 {my(T=M(10)); for(n=1, #T~, print(T[n, 1..n]))} \\ _Andrew Howroyd_, Dec 29 2019
%Y A330763 Column 1 is A000669.
%Y A330763 Main diagonal is A005804.
%Y A330763 Row sums are A330764.
%Y A330763 Cf. A330762 (leaves are multisets).
%K A330763 nonn,tabl
%O A330763 1,3
%A A330763 _Andrew Howroyd_, Dec 29 2019