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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 (list; graph; refs; listen; history; text; internal format)
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
CROSSREFS
Sequence in context: A359603 A255512 A039959 * A186335 A010363 A119561
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 18 2018
EXTENSIONS
Terms a(11) and beyond from Andrew Howroyd, Oct 26 2018
STATUS
approved

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Last modified March 29 10:59 EDT 2024. Contains 371277 sequences. (Running on oeis4.)