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A360956 Number of finite even-length multisets of positive integers whose right half sums to n.

%I A360956 #9 Mar 11 2023 14:21:26
%S A360956 1,1,3,5,10,13,26,31,55,73,112,140,233,276,405,539,750,931,1327,1627,
%T A360956 2259,2839,3708,4624,6237,7636,9823,12275,15715,19227,24735,30000,
%U A360956 37930,46339,57574,70374,87704,105606,129998,157417,193240,231769,283585,339052,411682,493260
%N A360956 Number of finite even-length multisets of positive integers whose right half sums to n.
%H A360956 Andrew Howroyd, <a href="/A360956/b360956.txt">Table of n, a(n) for n = 0..1000</a>
%F A360956 G.f.: 1 + Sum_{k>=1} x^k/((1 - x^k)^(k+1) * Product_{j=1..k-1} (1-x^j)). - _Andrew Howroyd_, Mar 11 2023
%e A360956 The a(1) = 1 through a(5) = 13 multisets:
%e A360956   {1,1}  {1,2}      {1,3}          {1,4}              {1,5}
%e A360956          {2,2}      {2,3}          {2,4}              {2,5}
%e A360956          {1,1,1,1}  {3,3}          {3,4}              {3,5}
%e A360956                     {1,1,1,2}      {4,4}              {4,5}
%e A360956                     {1,1,1,1,1,1}  {1,1,1,3}          {5,5}
%e A360956                                    {1,1,2,2}          {1,1,1,4}
%e A360956                                    {1,2,2,2}          {1,1,2,3}
%e A360956                                    {2,2,2,2}          {1,2,2,3}
%e A360956                                    {1,1,1,1,1,2}      {2,2,2,3}
%e A360956                                    {1,1,1,1,1,1,1,1}  {1,1,1,1,1,3}
%e A360956                                                       {1,1,1,1,2,2}
%e A360956                                                       {1,1,1,1,1,1,1,2}
%e A360956                                                       {1,1,1,1,1,1,1,1,1,1}
%e A360956 For example, the multiset y = {1,2,2,3} has right half {2,3}, with sum 5, so y is counted under a(5).
%t A360956 Table[Length[Select[Join@@IntegerPartitions/@Range[0,3*k], EvenQ[Length[#]]&&Total[Take[#,Length[#]/2]]==k&]],{k,0,15}]
%o A360956 (PARI) seq(n)={my(s=1 + O(x*x^n), p=s); for(k=1, n, s += p*x^k/(1-x^k + O(x*x^(n-k)))^(k+1); p /= 1 - x^k); Vec(s)} \\ _Andrew Howroyd_, Mar 11 2023
%Y A360956 This is the even-length case of A360671 and A360673.
%Y A360956 First for prime indices, second for partitions, third for prime factors:
%Y A360956 - A360676 gives left sum (exclusive), counted by A360672, product A361200.
%Y A360956 - A360677 gives right sum (exclusive), counted by A360675, product A361201.
%Y A360956 - A360678 gives left sum (inclusive), counted by A360675, product A347043.
%Y A360956 - A360679 gives right sum (inclusive), counted by A360672, product A347044.
%Y A360956 Cf. A000041, A360674, A360954, A360955.
%K A360956 nonn
%O A360956 0,3
%A A360956 _Gus Wiseman_, Mar 09 2023
%E A360956 Terms a(16) and beyond from _Andrew Howroyd_, Mar 11 2023