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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
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.
Sequence in context: A342212 A061304 A109442 * A025723 A173106 A022784
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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Last modified May 22 10:20 EDT 2024. Contains 372745 sequences. (Running on oeis4.)