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A168017
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Triangle read by rows in which row n lists the number of partitions of n into parts divisible by d, where d is a divisor of n listed in decreasing order.
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7
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1, 1, 2, 1, 3, 1, 2, 5, 1, 7, 1, 2, 3, 11, 1, 15, 1, 2, 5, 22, 1, 3, 30, 1, 2, 7, 42, 1, 56, 1, 2, 3, 5, 11, 77, 1, 101, 1, 2, 15, 135, 1, 3, 7, 176, 1, 2, 5, 22, 231, 1, 297, 1, 2, 3, 11, 30, 385, 1, 490, 1, 2, 5, 7, 42, 627, 1, 3, 15, 792, 1, 2, 56, 1002
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OFFSET
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1,3
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COMMENTS
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Positive values of triangle A168016.
The number of terms of row n is equal to the number of divisors of n: A000005(n).
Note that the last term of each row is the number of partitions of n: A000041(n).
Also, it appears that row n lists the partition numbers of the divisors of n. [From Omar E. Pol, Nov 23 2009]
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LINKS
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Alois P. Heinz, Rows n = 1..1400, flattened
O. E. Pol, Illustration of the partitions of n, for n = 1 .. 9
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EXAMPLE
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Consider row n=8: (1, 2, 5, 22). The divisors of 8 listed in decreasing order are 8, 4, 2, 1 (see A056538). Also, there is 1 partition of 8 into parts divisible by 8. Also, there are 2 partitions of 8 into parts divisible by 4: {(8), (4+4)}. Also, there are 5 partitions of 8 into parts divisible by 2: {(8), (6+2), (4+4), (4+2+2), (2+2+2+2)}. Finally, there are 22 partitions of 8 into parts divisible by 1, because A000041(8)=22. Then the row 8 is formed by 1, 2, 5, 22.
Triangle begins:
1;
1, 2;
1, 3;
1, 2, 5;
1, 7;
1, 2, 3, 11;
1, 15;
1, 2, 5, 22;
1, 3, 30;
1, 2, 7, 42;
1, 56;
1, 2, 3, 5, 11, 77;
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MAPLE
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with(numtheory):
b:= proc(n, i, d) option remember;
if n<0 then 0
elif n=0 then 1
elif i<1 then 0
else b(n, i-d, d) +b(n-i, i, d)
fi
end:
T:= proc(n) local l;
l:= sort([divisors(n)[]], `>`);
seq (b(n, n, l[i]), i=1..nops(l))
end:
seq (T(n), n=1..30); # Alois P. Heinz, Oct 21 2011
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CROSSREFS
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Row sums give A047968.
Cf. A000005, A000041, A056538, A135010, A138121, A168016, A168018, A168019, A168020, A168021.
Sequence in context: A036262 A169613 A176572 * A046924 A108415 A136644
Adjacent sequences: A168014 A168015 A168016 * A168018 A168019 A168020
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KEYWORD
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nonn,tabf
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AUTHOR
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Omar E. Pol, Nov 22 2009
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STATUS
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approved
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