A320664 Number of non-isomorphic multiset partitions of weight n with all parts of odd size.

1, 1, 2, 6, 12, 30, 82, 198, 533, 1459, 4039, 11634, 34095, 102520, 316456, 995709, 3215552, 10591412, 35633438, 122499429, 428988392, 1532929060, 5579867442, 20677066725, 78027003260, 299413756170, 1168536196157, 4635420192861, 18678567555721, 76451691937279, 317625507668759

Offset

0,3

Comments

Also the number of non-isomorphic multiset partitions of weight n in which each vertex appears an odd number of times.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..50

Example

Non-isomorphic representatives of the a(1) = 1 through a(4) = 12 multiset partitions with all parts of odd size:
  {{1}}  {{1},{1}}  {{1,1,1}}      {{1},{1,1,1}}
         {{1},{2}}  {{1,2,2}}      {{1},{1,2,2}}
                    {{1,2,3}}      {{1},{2,2,2}}
                    {{1},{1},{1}}  {{1},{2,3,3}}
                    {{1},{2},{2}}  {{1},{2,3,4}}
                    {{1},{2},{3}}  {{2},{1,2,2}}
                                   {{3},{1,2,3}}
                                   {{1},{1},{1},{1}}
                                   {{1},{1},{2},{2}}
                                   {{1},{2},{2},{2}}
                                   {{1},{2},{3},{3}}
                                   {{1},{2},{3},{4}}

Prog

(PARI) \\ See links in A339645 for combinatorial species functions.
seq(n)={my(A=symGroupSeries(n)); NumUnlabeledObjsSeq(sCartProd(sExp(A), sExp((A-subst(A, x, -x))/2)))} \\ Andrew Howroyd, Jan 17 2023
(PARI)
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
J(q, t, k, y)={1/prod(j=1, #q, my(s=q[j], g=gcd(s, t)); (1 + O(x*x^k) - y^(s/g)*x^(s*t/g))^g)}
K(q, t, k) = Vec(J(q, t, k, 1)-J(q, t, k, -1), -k)/2
a(n)={my(s=0); forpart(q=n, s+=permcount(q)*polcoef(exp(x*Ser(sum(t=1, n, K(q, t, n)/t))), n)); s/n!} \\ Andrew Howroyd, Jan 17 2023

Crossrefs

Cf. A001055, A001222, A007716, A298118, A300300, A300301, A318871, A320663, A320665.

Sequence in context: A095349 A291518 A291445 * A022916 A073949 A367336

Adjacent sequences: A320661 A320662 A320663 * A320665 A320666 A320667

Keyword

nonn

Author

Gus Wiseman, Oct 18 2018

Extensions

Terms a(11) and beyond from Andrew Howroyd, Jan 16 2023

Status

approved