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A330776 Triangle read by rows: T(n,k) is the number of balanced reduced multisystems of weight n with atoms colored using exactly k colors.

%I A330776 #9 Jan 09 2020 19:41:27
%S A330776 1,1,1,2,6,4,6,37,63,32,20,262,870,1064,436,90,2217,12633,27824,26330,
%T A330776 9012,468,21882,201654,710712,1163320,895608,262760,2910,249852,
%U A330776 3578610,18924846,47608000,61786254,40042128,10270696,20644,3245520,70539124,538018360,1950556400,3792461176,4070160416,2275829088,518277560
%N A330776 Triangle read by rows: T(n,k) is the number of balanced reduced multisystems of weight n with atoms colored using exactly k colors.
%C A330776 See A330655 for the definition of a balanced reduced multisystem.
%C A330776 A balanced reduced multisystem of weight n with atoms of k colors corresponds with a rooted tree with n leaves of k colors with all leaves at the same depth and at least one node at each level of the tree having more than one child. The final condition is needed to ensure that the number of such trees is finite.
%H A330776 Andrew Howroyd, <a href="/A330776/b330776.txt">Table of n, a(n) for n = 1..1275</a> (first 50 rows)
%e A330776 Triangle begins:
%e A330776     1;
%e A330776     1,     1;
%e A330776     2,     6,      4;
%e A330776     6,    37,     63,     32;
%e A330776    20,   262,    870,   1064,     436;
%e A330776    90,  2217,  12633,  27824,   26330,   9012;
%e A330776   468, 21882, 201654, 710712, 1163320, 895608, 262760;
%e A330776   ...
%e A330776 The T(3,2) = 6 balanced reduced multisystems are: {1,1,2}, {1,2,2}, {{1},{1,2}}, {{1},{2,2}}, {{2},{1,1}}, {{2},{1,2}}.
%o A330776 (PARI)
%o A330776 EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
%o A330776 R(n,k)={my(v=vector(n), u=vector(n)); v[1]=k; for(n=1, #v, u += v*sum(j=n, #v, (-1)^(j-n)*binomial(j-1,n-1)); v=EulerT(v)); u}
%o A330776 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 A330776 {my(T=M(10)); for(n=1, #T~, print(T[n, 1..n]))}
%Y A330776 Column 1 is A318813.
%Y A330776 Main diagonal is A005121.
%Y A330776 Row sums are A330655.
%K A330776 nonn,tabl
%O A330776 1,4
%A A330776 _Andrew Howroyd_, Dec 30 2019