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A318285
Number of non-isomorphic multiset partitions of a multiset whose multiplicities are the prime indices of n.
14
1, 1, 2, 2, 3, 4, 5, 3, 7, 7, 7, 9, 11, 12, 16, 5, 15, 17, 22, 16, 29, 19, 30, 16, 21, 30, 23, 29, 42, 52, 56, 7, 47, 45, 57, 43, 77, 67, 77, 31, 101, 98, 135, 47, 85, 97, 176, 29, 66, 64, 118, 77, 231, 69, 97, 57, 181, 139, 297, 137, 385, 195, 166, 11, 162, 171, 490, 118
OFFSET
1,3
LINKS
FORMULA
a(n) = A317791(A181821(n)).
EXAMPLE
Non-isomorphic representatives of the a(12) = 9 multiset partitions of {1,1,2,3}:
{{1,1,2,3}}
{{1},{1,2,3}}
{{2},{1,1,3}}
{{1,1},{2,3}}
{{1,2},{1,3}}
{{1},{1},{2,3}}
{{1},{2},{1,3}}
{{2},{3},{1,1}}
{{1},{1},{2},{3}}
PROG
(PARI) \\ See links in A339645 for combinatorial species functions.
sig(n)={my(f=factor(n), sig=vector(primepi(vecmax(f[, 1])))); for(i=1, #f~, sig[primepi(f[i, 1])]=f[i, 2]); sig}
C(sig)={my(n=sum(i=1, #sig, i*sig[i]), A=Vec(symGroupSeries(n)-1), B=O(x*x^n), c=prod(i=1, #sig, if(sig[i], sApplyCI(A[sig[i]], sig[i], A[i], i), 1))); polcoef(OgfSeries(sCartProd(c*x^n + B, sExp(x*Ser(A) + B))), n)}
a(n)={if(n==1, 1, C(sig(n)))} \\ Andrew Howroyd, Jan 17 2023
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 23 2018
EXTENSIONS
Terms a(31) and beyond from Andrew Howroyd, Jan 17 2023
STATUS
approved