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 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