%I #15 Mar 11 2023 15:07:42

%S 1,2,7,13,27,37,73,89,156,205,315,387,644,749,1104,1442,2015,2453,

%T 3529,4239,5926,7360,9624,11842,16115,19445,25084,31137,39911,48374,

%U 62559,75135,95263,115763,143749,174874,218614,261419,321991,388712,477439,569968,698493

%N Number of multisets of positive integers whose right half (exclusive) sums to n.

%H Andrew Howroyd, <a href="/A360673/b360673.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: 1 + Sum_{k>=1} x^k*(2 - x^k)/((1 - x^k)^(k+2) * Product_{j=1..k-1} (1-x^j)). - _Andrew Howroyd_, Mar 11 2023

%e The a(0) = 1 through a(3) = 13 multisets:

%e {} {1,1} {1,2} {1,3}

%e {1,1,1} {2,2} {2,3}

%e {1,1,2} {3,3}

%e {1,2,2} {1,1,3}

%e {2,2,2} {1,2,3}

%e {1,1,1,1} {1,3,3}

%e {1,1,1,1,1} {2,2,3}

%e {2,3,3}

%e {3,3,3}

%e {1,1,1,2}

%e {1,1,1,1,2}

%e {1,1,1,1,1,1}

%e {1,1,1,1,1,1,1}

%e For example, the multiset y = {1,1,1,1,2} has right half (exclusive) {1,2}, with sum 3, so y is counted under a(3).

%t Table[Length[Select[Join@@IntegerPartitions/@Range[0,3*k], Total[Take[#,Floor[Length[#]/2]]]==k&]],{k,0,15}]

%o (PARI) seq(n)={my(s=1 + O(x*x^n), p=s); for(k=1, n, s += p*x^k*(2-x^k)/(1-x^k + O(x*x^(n-k)))^(k+2); p /= 1 - x^k); Vec(s)} \\ _Andrew Howroyd_, Mar 11 2023

%Y The inclusive version is A360671.

%Y Column sums of A360672.

%Y The case of sets is A360954, inclusive A360955.

%Y The even-length case is A360956.

%Y A359893 and A359901 count partitions by median.

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

%Y - A360676 gives left sum (exclusive), counted by A360672, product A361200.

%Y - A360677 gives right sum (exclusive), counted by A360675, product A361201.

%Y - A360678 gives left sum (inclusive), counted by A360675, product A347043.

%Y - A360679 gives right sum (inclusive), counted by A360672, product A347044.

%Y Cf. A000041, A360616, A360617, A360674, A360675, A360953.

%K nonn

%O 0,2

%A _Gus Wiseman_, Mar 04 2023

%E Terms a(21) and beyond from _Andrew Howroyd_, Mar 11 2023