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 A360954 Number of finite sets of positive integers whose right half (exclusive) sums to n. 5
 1, 0, 1, 3, 6, 10, 15, 22, 29, 41, 50, 70, 81, 113, 126, 176, 191, 264, 286, 389, 413, 569, 595, 798, 861, 1121, 1187, 1585, 1653, 2132, 2334, 2906, 3111, 4006, 4234, 5252, 5818, 6995, 7620, 9453, 10102, 12165, 13663, 15940, 17498, 21127, 22961, 26881, 30222, 34678, 38569 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 LINKS Andrew Howroyd, Table of n, a(n) for n = 0..1000 FORMULA 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 EXAMPLE The a(2) = 1 through a(7) = 22 sets: {1,2} {1,3} {1,4} {1,5} {1,6} {1,7} {2,3} {2,4} {2,5} {2,6} {2,7} {1,2,3} {3,4} {3,5} {3,6} {3,7} {1,2,4} {4,5} {4,6} {4,7} {1,3,4} {1,2,5} {5,6} {5,7} {2,3,4} {1,3,5} {1,2,6} {6,7} {1,4,5} {1,3,6} {1,2,7} {2,3,5} {1,4,6} {1,3,7} {2,4,5} {1,5,6} {1,4,7} {3,4,5} {2,3,6} {1,5,7} {2,4,6} {1,6,7} {2,5,6} {2,3,7} {3,4,6} {2,4,7} {3,5,6} {2,5,7} {4,5,6} {2,6,7} {3,4,7} {3,5,7} {3,6,7} {4,5,7} {4,6,7} {5,6,7} {1,2,3,4} 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). MATHEMATICA Table[Length[Select[Join@@IntegerPartitions/@Range[0, 3*k], UnsameQ@@#&&Total[Take[#, Floor[Length[#]/2]]]==k&]], {k, 0, 15}] PROG (PARI) \\ P(n, k) is A072233(n, k). P(n, k)=polcoef(1/prod(k=1, k, 1 - x^k + O(x*x^n)), n) 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 CROSSREFS The version for multisets is A360673, inclusive A360671. The inclusive version is A360955. First for prime indices, second for partitions, third for prime factors: - A360676 gives left sum (exclusive), counted by A360672, product A361200. - A360677 gives right sum (exclusive), counted by A360675, product A361201. - A360678 gives left sum (inclusive), counted by A360675, product A347043. - A360679 gives right sum (inclusive), counted by A360672, product A347044. Cf. A000009, A072233, A359893, A359901, A360956. Sequence in context: A342212 A061304 A109442 * A025723 A173106 A022784 Adjacent sequences: A360951 A360952 A360953 * A360955 A360956 A360957 KEYWORD nonn AUTHOR Gus Wiseman, Mar 09 2023 EXTENSIONS Terms a(16) and beyond from Andrew Howroyd, Mar 13 2023 STATUS approved

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