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.

1, 1, 1, 2, 6, 4, 6, 37, 63, 32, 20, 262, 870, 1064, 436, 90, 2217, 12633, 27824, 26330, 9012, 468, 21882, 201654, 710712, 1163320, 895608, 262760, 2910, 249852, 3578610, 18924846, 47608000, 61786254, 40042128, 10270696, 20644, 3245520, 70539124, 538018360, 1950556400, 3792461176, 4070160416, 2275829088, 518277560

Offset: 1

Author

Andrew Howroyd, Dec 30 2019

Keywords

• nonn
• tabl

Comments

See A330655 for the definition of a balanced reduced multisystem.

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.

Examples

			Triangle begins:
    1;
    1,     1;
    2,     6,      4;
    6,    37,     63,     32;
   20,   262,    870,   1064,     436;
   90,  2217,  12633,  27824,   26330,   9012;
  468, 21882, 201654, 710712, 1163320, 895608, 262760;
  ...
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}}.
		

Links

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

Crossrefs

Column 1 is A318813.

Main diagonal is A005121.

Row sums are A330655.

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, u += v*sum(j=n, #v, (-1)^(j-n)*binomial(j-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]))}