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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

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%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