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A213943 Row sums of partition array A213939 and triangle A213940: number of representative bracelets with n beads and up to n colors.

%I A213943 #20 Jan 26 2025 14:12:58
%S A213943 1,2,3,9,28,144,832,6012,48447,444198,4469834,49650464,597810739,
%T A213943 7809600123,109524985564,1646900490716,26373465572350,448901183773766,
%U A213943 8083772124339442,153686286512223573,3074405841292532560,64582422678961767945
%N A213943 Row sums of partition array A213939 and triangle A213940: number of representative bracelets with n beads and up to n colors.
%C A213943 See A213939 for representative bracelets of a color class defined by a signature, given by a partition.
%C A213943 If color c[j] is written as j, for j from {1, 2, ... ,n}, the representative multisets, corresponding to the bracelets in question, are the ones with the least sum of their members.
%C A213943 E.g., n=4, m=3: signature [2,1,1] (partition of n with 4 parts), representative multiset (written as an ordered list by convention) [1,1,2,3], with the two representative bracelets 1123 and 1213, both taken cyclically.
%C A213943 Number of bracelets with n beads over a n-ary alphabet {a1,a2,...,an} such that #(w,a1) >= #(w,a2) >= ... >= #(w,ak) >= 0, where #(w,x) counts the letters x in word w. - _Andrew Howroyd_, Dec 21 2017
%H A213943 Andrew Howroyd, <a href="/A213943/b213943.txt">Table of n, a(n) for n = 1..200</a>
%F A213943 a(n) = sum(A213939(n,k),k=1..p(n)), with p(n)=A000041(n), n >= 1.
%F A213943 a(n) = sum(A213940(n,m),m=1..n), n >= 1.
%e A213943 The a(4)= 9 representative bracelets are (j for c[j]):  1111, 1112, 1122, 1212, 1123, 1213, 1234, 1324 and 1243, all taken cyclically.
%o A213943 (PARI)
%o A213943 a(n)={ if(n==0, 1,
%o A213943   my(p=serlaplace(prod(k=1, n, 1/(1-x^k/k!) + O(x*x^n))));
%o A213943   my(c=sumdiv(n, d, eulerphi(n/d)*polcoeff(p, d))/n);
%o A213943   my(r=if(n%2, sum(d=0, (n-1)/2, binomial((n-1)/2, d)*polcoeff(p, d)), polcoeff(p, n/2) + sum(d=0, n/2-1, binomial(n/2-1, d)*polcoeff(p, n/2-1-d)*(2^d + if(d%2, 0, binomial(d, d/2))))/2));
%o A213943   ( (c + r)/2 ) )
%o A213943 } \\ _Andrew Howroyd_, Dec 21 2017
%Y A213943 Row sums of A213940.
%Y A213943 Row sums of A214609.
%Y A213943 Cf. A072605 (representative necklaces).
%K A213943 nonn
%O A213943 1,2
%A A213943 _Wolfdieter Lang_, Jul 20 2012