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A320663
Number of non-isomorphic multiset partitions of weight n using singletons or pairs.
29
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
OFFSET
0,3
LINKS
EXAMPLE
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}}
PROG
(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
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 18 2018
EXTENSIONS
Terms a(11) and beyond from Andrew Howroyd, Oct 26 2018
STATUS
approved