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A330762 Triangle read by rows: T(n,k) is the number of series-reduced rooted trees whose leaves are multisets of colors with a total of n elements using exactly k colors.

%I A330762 #11 Jan 09 2020 19:40:21
%S A330762 1,2,2,4,12,8,11,67,114,58,30,376,1230,1496,612,96,2174,12038,26156,
%T A330762 24570,8374,308,12792,113028,389968,630300,481284,140408,1052,76972,
%U A330762 1043355,5363331,13259870,17008218,10930150,2785906,3648,471260,9574934,70524256,250201560,479284952,508811114,282141552,63830764
%N A330762 Triangle read by rows: T(n,k) is the number of series-reduced rooted trees whose leaves are multisets of colors with a total of n elements using exactly k colors.
%H A330762 Andrew Howroyd, <a href="/A330762/b330762.txt">Table of n, a(n) for n = 1..1275</a> (first 50 rows)
%e A330762 Triangle begins:
%e A330762      1;
%e A330762      2,     2;
%e A330762      4,    12,       8;
%e A330762     11,    67,     114,      58;
%e A330762     30,   376,    1230,    1496,      612;
%e A330762     96,  2174,   12038,   26156,    24570,     8374;
%e A330762    308, 12792,  113028,  389968,   630300,   481284,   140408;
%e A330762   1052, 76972, 1043355, 5363331, 13259870, 17008218, 10930150, 2785906;
%e A330762   ...
%e A330762 The T(3,2) = 12 trees are: (122), (112), ((1)(22)), ((1)(12)), ((2)(12)), ((2)(11)), ((1)(2)(2)), ((1)(1)(2)), ((1)((2)(2))), ((1)((1)(2))), ((2)((1)(2))), ((2)((1)(1))).
%o A330762 (PARI)
%o A330762 EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
%o A330762 R(n, k)={my(v=[]); for(n=1, n, v=concat(v, EulerT(concat(v, [binomial(n+k-1, k-1)]))[n])); v}
%o A330762 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 A330762 { my(T=M(10)); for(n=1, #T~, print(T[n, 1..n])) }
%Y A330762 Column 1 is A141268.
%Y A330762 Main diagonal is A005804.
%Y A330762 Row sums are A330469.
%Y A330762 Cf. A330763 (leaves are sets).
%K A330762 nonn,tabl
%O A330762 1,2
%A A330762 _Andrew Howroyd_, Dec 29 2019