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A360671
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Number of multisets whose right half (inclusive) sums to n.
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17
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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
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OFFSET
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0,2
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LINKS
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FORMULA
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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
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EXAMPLE
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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).
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MATHEMATICA
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Table[Length[Select[Join@@IntegerPartitions/@Range[0, 3*k], Total[Take[#, Ceiling[Length[#]/2]]]==k&]], {k, 0, 15}]
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PROG
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(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
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CROSSREFS
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Column sums of A360675 with rows reversed.
A360672 counts partitions by left sum (exclusive).
A360679 gives right sum (inclusive) of prime indices.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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