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A364909
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Number of ways to write n as a nonnegative linear combination of a strict integer composition of n.
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4
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1, 1, 1, 5, 5, 7, 51, 45, 89, 109, 709, 733, 1495, 1935, 3119, 13785, 16611, 29035, 44611, 68733, 95193, 372897, 435007, 781345, 1177181, 1866659, 2600537, 3906561, 12052631, 14610799, 25407653, 37652265, 59943351, 84060993, 128112805, 172172117, 480353257, 578740011
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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 (presumed 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(0) = 1 through a(5) = 7 ways:
. 1*1 1*2 1*3 1*4 1*5
0*2+3*1 0*3+4*1 0*4+5*1
1*1+1*2 1*1+1*3 1*1+1*4
1*2+1*1 1*3+1*1 1*2+1*3
3*1+0*2 4*1+0*3 1*3+1*2
1*4+1*1
5*1+0*4
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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[n, ptn], {ptn, Join@@Permutations/@Select[IntegerPartitions[n], UnsameQ@@#&]}]], {n, 0, 5}]
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PROG
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(Python)
from math import factorial
from sympy.utilities.iterables import partitions
if n == 0: return 1
aset = tuple(set(p) for p in partitions(n) if max(p.values(), default=0)==1)
return sum(factorial(len(t)) for p in partitions(n) for t in aset if set(p).issubset(t)) # Chai Wah Wu, Sep 21 2023
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CROSSREFS
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The case with no zero coefficients is A032020.
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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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