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.
%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