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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.
3
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
1,4
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.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)
EXAMPLE
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}}.
PROG
(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]))}
CROSSREFS
Column 1 is A318813.
Main diagonal is A005121.
Row sums are A330655.
Sequence in context: A151689 A216833 A242046 * A327458 A202347 A364897
KEYWORD
nonn,tabl
AUTHOR
Andrew Howroyd, Dec 30 2019
STATUS
approved