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

%I A320664 #14 Jan 17 2023 12:08:43
%S A320664 1,1,2,6,12,30,82,198,533,1459,4039,11634,34095,102520,316456,995709,
%T A320664 3215552,10591412,35633438,122499429,428988392,1532929060,5579867442,
%U A320664 20677066725,78027003260,299413756170,1168536196157,4635420192861,18678567555721,76451691937279,317625507668759
%N A320664 Number of non-isomorphic multiset partitions of weight n with all parts of odd size.
%C A320664 Also the number of non-isomorphic multiset partitions of weight n in which each vertex appears an odd number of times.
%H A320664 Andrew Howroyd, <a href="/A320664/b320664.txt">Table of n, a(n) for n = 0..50</a>
%e A320664 Non-isomorphic representatives of the a(1) = 1 through a(4) = 12 multiset partitions with all parts of odd size:
%e A320664   {{1}}  {{1},{1}}  {{1,1,1}}      {{1},{1,1,1}}
%e A320664          {{1},{2}}  {{1,2,2}}      {{1},{1,2,2}}
%e A320664                     {{1,2,3}}      {{1},{2,2,2}}
%e A320664                     {{1},{1},{1}}  {{1},{2,3,3}}
%e A320664                     {{1},{2},{2}}  {{1},{2,3,4}}
%e A320664                     {{1},{2},{3}}  {{2},{1,2,2}}
%e A320664                                    {{3},{1,2,3}}
%e A320664                                    {{1},{1},{1},{1}}
%e A320664                                    {{1},{1},{2},{2}}
%e A320664                                    {{1},{2},{2},{2}}
%e A320664                                    {{1},{2},{3},{3}}
%e A320664                                    {{1},{2},{3},{4}}
%o A320664 (PARI) \\ See links in A339645 for combinatorial species functions.
%o A320664 seq(n)={my(A=symGroupSeries(n)); NumUnlabeledObjsSeq(sCartProd(sExp(A), sExp((A-subst(A,x,-x))/2)))} \\ _Andrew Howroyd_, Jan 17 2023
%o A320664 (PARI)
%o A320664 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 A320664 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 A320664 K(q, t, k) = Vec(J(q,t,k,1)-J(q,t,k,-1), -k)/2
%o A320664 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 A320664 Cf. A001055, A001222, A007716, A298118, A300300, A300301, A318871, A320663, A320665.
%K A320664 nonn
%O A320664 0,3
%A A320664 _Gus Wiseman_, Oct 18 2018
%E A320664 Terms a(11) and beyond from _Andrew Howroyd_, Jan 16 2023