cp's OEIS Frontend

A318285 Number of non-isomorphic multiset partitions of a multiset whose multiplicities are the prime indices of n.

%I A318285 #12 Jan 17 2023 21:24:38
%S A318285 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,
%T A318285 29,42,52,56,7,47,45,57,43,77,67,77,31,101,98,135,47,85,97,176,29,66,
%U A318285 64,118,77,231,69,97,57,181,139,297,137,385,195,166,11,162,171,490,118
%N A318285 Number of non-isomorphic multiset partitions of a multiset whose multiplicities are the prime indices of n.
%H A318285 Andrew Howroyd, <a href="/A318285/b318285.txt">Table of n, a(n) for n = 1..130</a>
%F A318285 a(n) = A317791(A181821(n)).
%e A318285 Non-isomorphic representatives of the a(12) = 9 multiset partitions of {1,1,2,3}:
%e A318285   {{1,1,2,3}}
%e A318285   {{1},{1,2,3}}
%e A318285   {{2},{1,1,3}}
%e A318285   {{1,1},{2,3}}
%e A318285   {{1,2},{1,3}}
%e A318285   {{1},{1},{2,3}}
%e A318285   {{1},{2},{1,3}}
%e A318285   {{2},{3},{1,1}}
%e A318285   {{1},{1},{2},{3}}
%o A318285 (PARI) \\ See links in A339645 for combinatorial species functions.
%o A318285 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}
%o A318285 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)}
%o A318285 a(n)={if(n==1, 1, C(sig(n)))} \\ _Andrew Howroyd_, Jan 17 2023
%Y A318285 Cf. A007716, A181821, A255906, A269134, A305936, A317533, A317791.
%Y A318285 Cf. A318283, A318284, A318287, A318362, A318371.
%K A318285 nonn
%O A318285 1,3
%A A318285 _Gus Wiseman_, Aug 23 2018
%E A318285 Terms a(31) and beyond from _Andrew Howroyd_, Jan 17 2023