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A360954 Number of finite sets of positive integers whose right half (exclusive) sums to n.

%I A360954 #9 Mar 13 2023 17:47:56
%S A360954 1,0,1,3,6,10,15,22,29,41,50,70,81,113,126,176,191,264,286,389,413,
%T A360954 569,595,798,861,1121,1187,1585,1653,2132,2334,2906,3111,4006,4234,
%U A360954 5252,5818,6995,7620,9453,10102,12165,13663,15940,17498,21127,22961,26881,30222,34678,38569
%N A360954 Number of finite sets of positive integers whose right half (exclusive) sums to n.
%H A360954 Andrew Howroyd, <a href="/A360954/b360954.txt">Table of n, a(n) for n = 0..1000</a>
%F A360954 a(n) = Sum_{w>=1} Sum_{h=w+1..floor((n-binomial(w,2))/w)} binomial(h,w+1) * A072233(n - w*h - binomial(w,2), w-1) for n > 0. - _Andrew Howroyd_, Mar 13 2023
%e A360954 The a(2) = 1 through a(7) = 22 sets:
%e A360954   {1,2}  {1,3}    {1,4}    {1,5}    {1,6}    {1,7}
%e A360954          {2,3}    {2,4}    {2,5}    {2,6}    {2,7}
%e A360954          {1,2,3}  {3,4}    {3,5}    {3,6}    {3,7}
%e A360954                   {1,2,4}  {4,5}    {4,6}    {4,7}
%e A360954                   {1,3,4}  {1,2,5}  {5,6}    {5,7}
%e A360954                   {2,3,4}  {1,3,5}  {1,2,6}  {6,7}
%e A360954                            {1,4,5}  {1,3,6}  {1,2,7}
%e A360954                            {2,3,5}  {1,4,6}  {1,3,7}
%e A360954                            {2,4,5}  {1,5,6}  {1,4,7}
%e A360954                            {3,4,5}  {2,3,6}  {1,5,7}
%e A360954                                     {2,4,6}  {1,6,7}
%e A360954                                     {2,5,6}  {2,3,7}
%e A360954                                     {3,4,6}  {2,4,7}
%e A360954                                     {3,5,6}  {2,5,7}
%e A360954                                     {4,5,6}  {2,6,7}
%e A360954                                              {3,4,7}
%e A360954                                              {3,5,7}
%e A360954                                              {3,6,7}
%e A360954                                              {4,5,7}
%e A360954                                              {4,6,7}
%e A360954                                              {5,6,7}
%e A360954                                              {1,2,3,4}
%e A360954 For example, the set y = {1,2,3,4} has right half (exclusive) {3,4}, with sum 7, so y is counted under a(7).
%t A360954 Table[Length[Select[Join@@IntegerPartitions/@Range[0,3*k],UnsameQ@@#&&Total[Take[#,Floor[Length[#]/2]]]==k&]],{k,0,15}]
%o A360954 (PARI) \\ P(n,k) is A072233(n,k).
%o A360954 P(n,k)=polcoef(1/prod(k=1, k, 1 - x^k + O(x*x^n)), n)
%o A360954 a(n)=if(n==0, 1, sum(w=1, sqrt(n), my(t=binomial(w,2)); sum(h=w+1, (n-t)\w, binomial(h, w+1) * P(n-w*h-t, w-1)))) \\ _Andrew Howroyd_, Mar 13 2023
%Y A360954 The version for multisets is A360673, inclusive A360671.
%Y A360954 The inclusive version is A360955.
%Y A360954 First for prime indices, second for partitions, third for prime factors:
%Y A360954 - A360676 gives left sum (exclusive), counted by A360672, product A361200.
%Y A360954 - A360677 gives right sum (exclusive), counted by A360675, product A361201.
%Y A360954 - A360678 gives left sum (inclusive), counted by A360675, product A347043.
%Y A360954 - A360679 gives right sum (inclusive), counted by A360672, product A347044.
%Y A360954 Cf. A000009, A072233, A359893, A359901, A360956.
%K A360954 nonn
%O A360954 0,4
%A A360954 _Gus Wiseman_, Mar 09 2023
%E A360954 Terms a(16) and beyond from _Andrew Howroyd_, Mar 13 2023