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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 (list; table; graph; refs; listen; history; text; internal format)
 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 A266120 Adjacent sequences:  A330773 A330774 A330775 * A330777 A330778 A330779 KEYWORD nonn,tabl AUTHOR Andrew Howroyd, Dec 30 2019 STATUS approved

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Last modified September 22 13:36 EDT 2020. Contains 337289 sequences. (Running on oeis4.)