

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
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OFFSET

0,4


LINKS



FORMULA

a(n) = Sum_{w>=1} Sum_{h=w+1..floor((nbinomial(w,2))/w)} binomial(h,w+1) * A072233(n  w*h  binomial(w,2), w1) 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, (nt)\w, binomial(h, w+1) * P(nw*ht, w1)))) \\ Andrew Howroyd, Mar 13 2023


CROSSREFS

First for prime indices, second for partitions, third for prime factors:


KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



