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%I #9 Oct 26 2018 20:36:02
%S 1,1,4,7,21,40,106,216,534,1139,2715,5962,14012,31420,73484,167617,
%T 392714,908600,2140429,5015655,11905145,28228533,67590229,162067916,
%U 391695348,949359190,2316618809,5673557284,13979155798,34583650498,86034613145,214948212879
%N Number of non-isomorphic multiset partitions of weight n using singletons or pairs.
%H Andrew Howroyd, <a href="/A320663/b320663.txt">Table of n, a(n) for n = 0..50</a>
%e Non-isomorphic representatives of the a(1) = 1 through a(4) = 21 multiset partitions:
%e {{1}} {{1,1}} {{1},{1,1}} {{1,1},{1,1}}
%e {{1,2}} {{1},{2,2}} {{1,1},{2,2}}
%e {{1},{1}} {{1},{2,3}} {{1,2},{1,2}}
%e {{1},{2}} {{2},{1,2}} {{1,2},{2,2}}
%e {{1},{1},{1}} {{1,2},{3,3}}
%e {{1},{2},{2}} {{1,2},{3,4}}
%e {{1},{2},{3}} {{1,3},{2,3}}
%e {{1},{1},{1,1}}
%e {{1},{1},{2,2}}
%e {{1},{1},{2,3}}
%e {{1},{2},{1,2}}
%e {{1},{2},{2,2}}
%e {{1},{2},{3,3}}
%e {{1},{2},{3,4}}
%e {{1},{3},{2,3}}
%e {{2},{2},{1,2}}
%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) EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
%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 gs(v) = {sum(i=2, #v, sum(j=1, i-1, my(g=gcd(v[i],v[j])); g*x^(2*v[i]*v[j]/g))) + sum(i=1, #v, my(r=v[i]); (1 + (1+r)%2)*x^r + ((1+r)\2)*x^(2*r))}
%o a(n)={my(s=0); forpart(p=n, s+=permcount(p)*EulerT(Vec(gs(p) + O(x*x^n), -n))[n]); s/n!} \\ _Andrew Howroyd_, Oct 26 2018
%Y Cf. A001055, A001222, A001358, A005117, A006881, A007716, A007717, A037143, A320462, A320655, A320656, A320664, A320665.
%K nonn
%O 0,3
%A _Gus Wiseman_, Oct 18 2018
%E Terms a(11) and beyond from _Andrew Howroyd_, Oct 26 2018
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