cp's OEIS Frontend

A320663 Number of non-isomorphic multiset partitions of weight n using singletons or pairs.

%I A320663 #9 Oct 26 2018 20:36:02
%S A320663 1,1,4,7,21,40,106,216,534,1139,2715,5962,14012,31420,73484,167617,
%T A320663 392714,908600,2140429,5015655,11905145,28228533,67590229,162067916,
%U A320663 391695348,949359190,2316618809,5673557284,13979155798,34583650498,86034613145,214948212879
%N A320663 Number of non-isomorphic multiset partitions of weight n using singletons or pairs.
%H A320663 Andrew Howroyd, <a href="/A320663/b320663.txt">Table of n, a(n) for n = 0..50</a>
%e A320663 Non-isomorphic representatives of the a(1) = 1 through a(4) = 21 multiset partitions:
%e A320663   {{1}}  {{1,1}}    {{1},{1,1}}    {{1,1},{1,1}}
%e A320663          {{1,2}}    {{1},{2,2}}    {{1,1},{2,2}}
%e A320663          {{1},{1}}  {{1},{2,3}}    {{1,2},{1,2}}
%e A320663          {{1},{2}}  {{2},{1,2}}    {{1,2},{2,2}}
%e A320663                     {{1},{1},{1}}  {{1,2},{3,3}}
%e A320663                     {{1},{2},{2}}  {{1,2},{3,4}}
%e A320663                     {{1},{2},{3}}  {{1,3},{2,3}}
%e A320663                                    {{1},{1},{1,1}}
%e A320663                                    {{1},{1},{2,2}}
%e A320663                                    {{1},{1},{2,3}}
%e A320663                                    {{1},{2},{1,2}}
%e A320663                                    {{1},{2},{2,2}}
%e A320663                                    {{1},{2},{3,3}}
%e A320663                                    {{1},{2},{3,4}}
%e A320663                                    {{1},{3},{2,3}}
%e A320663                                    {{2},{2},{1,2}}
%e A320663                                    {{1},{1},{1},{1}}
%e A320663                                    {{1},{1},{2},{2}}
%e A320663                                    {{1},{2},{2},{2}}
%e A320663                                    {{1},{2},{3},{3}}
%e A320663                                    {{1},{2},{3},{4}}
%o A320663 (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
%o A320663 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}
%o A320663 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))}
%o A320663 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
%Y A320663 Cf. A001055, A001222, A001358, A005117, A006881, A007716, A007717, A037143, A320462, A320655, A320656, A320664, A320665.
%K A320663 nonn
%O A320663 0,3
%A A320663 _Gus Wiseman_, Oct 18 2018
%E A320663 Terms a(11) and beyond from _Andrew Howroyd_, Oct 26 2018