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A365005
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Number of ways to write 2 as a nonnegative linear combination of a strict integer partition of n.
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1
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0, 1, 1, 2, 1, 2, 4, 4, 5, 6, 9, 10, 13, 15, 19, 23, 28, 33, 40, 47, 56, 67, 78, 92, 108, 126, 146, 171, 198, 229, 264, 305, 350, 403, 460, 527, 603, 687, 781, 889, 1009, 1144, 1295, 1464, 1653, 1866, 2101, 2364, 2659, 2984, 3347, 3752, 4200, 4696, 5248, 5858
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OFFSET
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0,4
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COMMENTS
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A way of writing n as a (nonnegative) linear combination of a finite sequence y is any sequence of pairs (k_i,y_i) such that k_i >= 0 and Sum k_i*y_i = n. For example, the pairs ((3,1),(1,1),(1,1),(0,2)) are a way of writing 5 as a linear combination of (1,1,1,2), namely 5 = 3*1 + 1*1 + 1*1 + 0*2. Of course, there are A000041(n) ways to write n as a linear combination of (1..n).
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LINKS
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EXAMPLE
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The a(6) = 4 ways:
0*5 + 2*1
0*4 + 1*2
0*3 + 0*2 + 2*1
0*3 + 1*2 + 0*1
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MATHEMATICA
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combs[n_, y_]:=With[{s=Table[{k, i}, {k, y}, {i, 0, Floor[n/k]}]}, Select[Tuples[s], Total[Times@@@#]==n&]];
Table[Length[Join@@Table[combs[2, ptn], {ptn, Select[IntegerPartitions[n], UnsameQ@@#&]}]], {n, 0, 30}]
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CROSSREFS
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For 1 instead of 2 we have A096765.
A364350 counts combination-free strict partitions, complement A364839.
A364913 counts combination-full partitions.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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