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.

1, 1, 1, 1, 4, 3, 2, 17, 33, 18, 5, 86, 321, 420, 180, 16, 520, 3306, 7752, 7650, 2700, 61, 3682, 37533, 140172, 238560, 189000, 56700, 272, 30050, 473604, 2644356, 6899070, 9196740, 6085800, 1587600, 1385, 278414, 6630909, 53244180, 199775820, 398328480, 435954960, 247665600, 57153600

Offset: 1

Author

Andrew Howroyd, Dec 30 2019

Keywords

• nonn
• tabl

Examples

			Triangle begins:
    1;
    1,     1;
    1,     4,      3;
    2,    17,     33,      18;
    5,    86,    321,     420,     180;
   16,   520,   3306,    7752,    7650,    2700;
   61,  3682,  37533,  140172,  238560,  189000,   56700;
  272, 30050, 473604, 2644356, 6899070, 9196740, 6085800, 1587600;
  ...
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)

Crossrefs

Column 1 is A000111.

Main diagonal is A006472.

Row sums are A330676.

Cf. A330776.

Programs

PARI

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
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}
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])))}
{my(T=M(10)); for(n=1, #T~, print(T[n, 1..n]))}