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A320663
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Number of non-isomorphic multiset partitions of weight n using singletons or pairs.
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29
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1, 1, 4, 7, 21, 40, 106, 216, 534, 1139, 2715, 5962, 14012, 31420, 73484, 167617, 392714, 908600, 2140429, 5015655, 11905145, 28228533, 67590229, 162067916, 391695348, 949359190, 2316618809, 5673557284, 13979155798, 34583650498, 86034613145, 214948212879
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OFFSET
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0,3
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LINKS
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EXAMPLE
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Non-isomorphic representatives of the a(1) = 1 through a(4) = 21 multiset partitions:
{{1}} {{1,1}} {{1},{1,1}} {{1,1},{1,1}}
{{1,2}} {{1},{2,2}} {{1,1},{2,2}}
{{1},{1}} {{1},{2,3}} {{1,2},{1,2}}
{{1},{2}} {{2},{1,2}} {{1,2},{2,2}}
{{1},{1},{1}} {{1,2},{3,3}}
{{1},{2},{2}} {{1,2},{3,4}}
{{1},{2},{3}} {{1,3},{2,3}}
{{1},{1},{1,1}}
{{1},{1},{2,2}}
{{1},{1},{2,3}}
{{1},{2},{1,2}}
{{1},{2},{2,2}}
{{1},{2},{3,3}}
{{1},{2},{3,4}}
{{1},{3},{2,3}}
{{2},{2},{1,2}}
{{1},{1},{1},{1}}
{{1},{1},{2},{2}}
{{1},{2},{2},{2}}
{{1},{2},{3},{3}}
{{1},{2},{3},{4}}
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PROG
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(PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
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}
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))}
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
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CROSSREFS
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Cf. A001055, A001222, A001358, A005117, A006881, A007716, A007717, A037143, A320462, A320655, A320656, A320664, A320665.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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