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A330778 Triangle read by rows: T(n,k) is the number of balanced reduced multisystems of weight n with maximum depth and atoms colored using exactly k colors.

%I A330778 #9 Jan 09 2020 19:42:00
%S A330778 1,1,1,1,4,3,2,17,33,18,5,86,321,420,180,16,520,3306,7752,7650,2700,
%T A330778 61,3682,37533,140172,238560,189000,56700,272,30050,473604,2644356,
%U A330778 6899070,9196740,6085800,1587600,1385,278414,6630909,53244180,199775820,398328480,435954960,247665600,57153600
%N A330778 Triangle read by rows: T(n,k) is the number of balanced reduced multisystems of weight n with maximum depth and atoms colored using exactly k colors.
%H A330778 Andrew Howroyd, <a href="/A330778/b330778.txt">Table of n, a(n) for n = 1..1275</a> (first 50 rows)
%e A330778 Triangle begins:
%e A330778     1;
%e A330778     1,     1;
%e A330778     1,     4,      3;
%e A330778     2,    17,     33,      18;
%e A330778     5,    86,    321,     420,     180;
%e A330778    16,   520,   3306,    7752,    7650,    2700;
%e A330778    61,  3682,  37533,  140172,  238560,  189000,   56700;
%e A330778   272, 30050, 473604, 2644356, 6899070, 9196740, 6085800, 1587600;
%e A330778   ...
%o A330778 (PARI)
%o A330778 EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
%o A330778 R(n, k)={my(v=vector(n), u=vector(n)); v[1]=k; for(n=1, #v, for(i=n, #v, u[i] += v[i]*(-1)^(i-n)*binomial(i-1, n-1)); v=EulerT(v)); u}
%o A330778 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 A330778 {my(T=M(10)); for(n=1, #T~, print(T[n, 1..n]))}
%Y A330778 Column 1 is A000111.
%Y A330778 Main diagonal is A006472.
%Y A330778 Row sums are A330676.
%Y A330778 Cf. A330776.
%K A330778 nonn,tabl
%O A330778 1,5
%A A330778 _Andrew Howroyd_, Dec 30 2019