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A339888
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Number of non-isomorphic multiset partitions of weight n into singletons or strict pairs.
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15
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1, 1, 3, 5, 13, 23, 55, 104, 236, 470, 1039, 2140, 4712, 9962, 21961, 47484, 105464, 232324, 521338, 1167825, 2651453, 6031136, 13863054, 31987058, 74448415, 174109134, 410265423, 971839195, 2317827540, 5558092098, 13412360692, 32542049038, 79424450486
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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) = 13 multiset partitions:
{{1}} {{1,2}} {{1},{2,3}} {{1,2},{1,2}}
{{1},{1}} {{2},{1,2}} {{1,2},{3,4}}
{{1},{2}} {{1},{1},{1}} {{1,3},{2,3}}
{{1},{2},{2}} {{1},{1},{2,3}}
{{1},{2},{3}} {{1},{2},{1,2}}
{{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 + ((r-1)\2)*x^(2*r))}
a(n)={if(n==0, 1, my(s=0); forpart(p=n, s+=permcount(p)*EulerT(Vec(gs(p) + O(x*x^n), -n))[n]); s/n!)} \\ Andrew Howroyd, Apr 16 2021
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CROSSREFS
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The version for set partitions is A000085, with ordered version A080599.
Non-isomorphic multiset partitions are counted by A007716.
The case without singletons is A007717.
The version allowing non-strict pairs (x,x) is A320663.
A339742 counts factorizations into distinct primes or squarefree semiprimes.
A339887 counts factorizations into primes or squarefree semiprimes.
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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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