A320664 - OEIS

%I #14 Jan 17 2023 12:08:43

%S 1,1,2,6,12,30,82,198,533,1459,4039,11634,34095,102520,316456,995709,

%T 3215552,10591412,35633438,122499429,428988392,1532929060,5579867442,

%U 20677066725,78027003260,299413756170,1168536196157,4635420192861,18678567555721,76451691937279,317625507668759

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

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

%H Andrew Howroyd, <a href="/A320664/b320664.txt">Table of n, a(n) for n = 0..50</a>

%e Non-isomorphic representatives of the a(1) = 1 through a(4) = 12 multiset partitions with all parts of odd size:

%e   {{1}}  {{1},{1}}  {{1,1,1}}      {{1},{1,1,1}}

%e          {{1},{2}}  {{1,2,2}}      {{1},{1,2,2}}

%e                     {{1,2,3}}      {{1},{2,2,2}}

%e                     {{1},{1},{1}}  {{1},{2,3,3}}

%e                     {{1},{2},{2}}  {{1},{2,3,4}}

%e                     {{1},{2},{3}}  {{2},{1,2,2}}

%e                                    {{3},{1,2,3}}

%e                                    {{1},{1},{1},{1}}

%e                                    {{1},{1},{2},{2}}

%e                                    {{1},{2},{2},{2}}

%e                                    {{1},{2},{3},{3}}

%e                                    {{1},{2},{3},{4}}

%o (PARI) \\ See links in A339645 for combinatorial species functions.

%o seq(n)={my(A=symGroupSeries(n)); NumUnlabeledObjsSeq(sCartProd(sExp(A), sExp((A-subst(A,x,-x))/2)))} \\ _Andrew Howroyd_, Jan 17 2023

%o (PARI)

%o 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}

%o 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)}

%o K(q, t, k) = Vec(J(q,t,k,1)-J(q,t,k,-1), -k)/2

%o 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

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

%K nonn

%O 0,3

%A _Gus Wiseman_, Oct 18 2018

%E Terms a(11) and beyond from _Andrew Howroyd_, Jan 16 2023