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A360671
Number of multisets whose right half (inclusive) sums to n.
17
1, 2, 5, 8, 16, 21, 42, 51, 90, 121, 185, 235, 386, 465, 679, 908, 1261, 1580, 2238, 2770, 3827, 4831, 6314, 7910, 10619, 13074, 16813, 21049, 26934, 33072, 42445, 51679, 65264, 79902, 99309, 121548, 151325, 182697, 224873, 272625, 334536, 401999, 491560, 588723
OFFSET
0,2
LINKS
FORMULA
G.f.: 1 + Sum_{k>=1} x^k*(2 - x^k)/((1 - x^k)^(k+1) * Product_{j=1..k-1} (1-x^j)). - Andrew Howroyd, Mar 11 2023
EXAMPLE
The a(0) = 1 through a(4) = 16 multisets:
{} {1} {2} {3} {4}
{1,1} {1,2} {1,3} {1,4}
{2,2} {2,3} {2,4}
{1,1,1} {3,3} {3,4}
{1,1,1,1} {1,1,2} {4,4}
{1,1,1,2} {1,1,3}
{1,1,1,1,1} {1,2,2}
{1,1,1,1,1,1} {2,2,2}
{1,1,1,3}
{1,1,2,2}
{1,2,2,2}
{2,2,2,2}
{1,1,1,1,2}
{1,1,1,1,1,2}
{1,1,1,1,1,1,1}
{1,1,1,1,1,1,1,1}
For example, the multiset y = {1,1,1,1,2} has right half (inclusive) {1,1,2}, with sum 4, so y is counted under a(4).
MATHEMATICA
Table[Length[Select[Join@@IntegerPartitions/@Range[0, 3*k], Total[Take[#, Ceiling[Length[#]/2]]]==k&]], {k, 0, 15}]
PROG
(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+1); p /= 1 - x^k); Vec(s)} \\ Andrew Howroyd, Mar 11 2023
CROSSREFS
The exclusive version is A360673.
Column sums of A360675 with rows reversed.
The case of sets is A360955, exclusive A360954.
The even-length case is A360956.
A360672 counts partitions by left sum (exclusive).
A360679 gives right sum (inclusive) of prime indices.
Sequence in context: A080084 A065093 A361270 * A168470 A295998 A129299
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 09 2023
EXTENSIONS
Terms a(24) and beyond from Andrew Howroyd, Mar 11 2023
STATUS
approved