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A360671 Number of multisets whose right half (inclusive) sums to n.

%I A360671 #9 Mar 11 2023 15:07:46
%S A360671 1,2,5,8,16,21,42,51,90,121,185,235,386,465,679,908,1261,1580,2238,
%T A360671 2770,3827,4831,6314,7910,10619,13074,16813,21049,26934,33072,42445,
%U A360671 51679,65264,79902,99309,121548,151325,182697,224873,272625,334536,401999,491560,588723
%N A360671 Number of multisets whose right half (inclusive) sums to n.
%H A360671 Andrew Howroyd, <a href="/A360671/b360671.txt">Table of n, a(n) for n = 0..1000</a>
%F A360671 G.f.: 1 + Sum_{k>=1} x^k*(2 - x^k)/((1 - x^k)^(k+1) * Product_{j=1..k-1} (1-x^j)). - _Andrew Howroyd_, Mar 11 2023
%e A360671 The a(0) = 1 through a(4) = 16 multisets:
%e A360671   {}  {1}    {2}        {3}            {4}
%e A360671       {1,1}  {1,2}      {1,3}          {1,4}
%e A360671              {2,2}      {2,3}          {2,4}
%e A360671              {1,1,1}    {3,3}          {3,4}
%e A360671              {1,1,1,1}  {1,1,2}        {4,4}
%e A360671                         {1,1,1,2}      {1,1,3}
%e A360671                         {1,1,1,1,1}    {1,2,2}
%e A360671                         {1,1,1,1,1,1}  {2,2,2}
%e A360671                                        {1,1,1,3}
%e A360671                                        {1,1,2,2}
%e A360671                                        {1,2,2,2}
%e A360671                                        {2,2,2,2}
%e A360671                                        {1,1,1,1,2}
%e A360671                                        {1,1,1,1,1,2}
%e A360671                                        {1,1,1,1,1,1,1}
%e A360671                                        {1,1,1,1,1,1,1,1}
%e A360671 For example, the multiset y = {1,1,1,1,2} has right half (inclusive) {1,1,2}, with sum 4, so y is counted under a(4).
%t A360671 Table[Length[Select[Join@@IntegerPartitions/@Range[0,3*k], Total[Take[#,Ceiling[Length[#]/2]]]==k&]],{k,0,15}]
%o A360671 (PARI) seq(n)={my(s=1 + O(x*x^n), p=s); for(k=1, n, s += p*x^k*(2-x^k)/(1-x^k + O(x*x^(n-k)))^(k+1); p /= 1 - x^k); Vec(s)} \\ _Andrew Howroyd_, Mar 11 2023
%Y A360671 The exclusive version is A360673.
%Y A360671 Column sums of A360675 with rows reversed.
%Y A360671 The case of sets is A360955, exclusive A360954.
%Y A360671 The even-length case is A360956.
%Y A360671 A360672 counts partitions by left sum (exclusive).
%Y A360671 A360679 gives right sum (inclusive) of prime indices.
%Y A360671 Cf. A000041, A347044, A361200, A361201.
%K A360671 nonn
%O A360671 0,2
%A A360671 _Gus Wiseman_, Mar 09 2023
%E A360671 Terms a(24) and beyond from _Andrew Howroyd_, Mar 11 2023