A330778 - OEIS

%I #9 Jan 09 2020 19:42:00

%S 1,1,1,1,4,3,2,17,33,18,5,86,321,420,180,16,520,3306,7752,7650,2700,

%T 61,3682,37533,140172,238560,189000,56700,272,30050,473604,2644356,

%U 6899070,9196740,6085800,1587600,1385,278414,6630909,53244180,199775820,398328480,435954960,247665600,57153600

%N 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 Andrew Howroyd, <a href="/A330778/b330778.txt">Table of n, a(n) for n = 1..1275</a> (first 50 rows)

%e Triangle begins:

%e     1;

%e     1,     1;

%e     1,     4,      3;

%e     2,    17,     33,      18;

%e     5,    86,    321,     420,     180;

%e    16,   520,   3306,    7752,    7650,    2700;

%e    61,  3682,  37533,  140172,  238560,  189000,   56700;

%e   272, 30050, 473604, 2644356, 6899070, 9196740, 6085800, 1587600;

%e   ...

%o (PARI)

%o EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}

%o 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 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 {my(T=M(10)); for(n=1, #T~, print(T[n, 1..n]))}

%Y Column 1 is A000111.

%Y Main diagonal is A006472.

%Y Row sums are A330676.

%Y Cf. A330776.

%K nonn,tabl

%O 1,5

%A _Andrew Howroyd_, Dec 30 2019