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A168021
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Triangle T(n,k) read by rows in which row n lists the number of partitions of n into parts divisible by k.
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16
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1, 2, 1, 3, 0, 1, 5, 2, 0, 1, 7, 0, 0, 0, 1, 11, 3, 2, 0, 0, 1, 15, 0, 0, 0, 0, 0, 1, 22, 5, 0, 2, 0, 0, 0, 1, 30, 0, 3, 0, 0, 0, 0, 0, 1, 42, 7, 0, 0, 2, 0, 0, 0, 0, 1, 56, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 77, 11, 5, 3, 0, 2, 0, 0, 0, 0, 0, 1
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OFFSET
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1,2
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COMMENTS
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The row-reversed version is A168016.
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LINKS
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FORMULA
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T(n,k) = A000041(n/k) if k|n, else T(n,k)=0.
T(2*n-1, n+1) = A000007(n-2). (End)
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EXAMPLE
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Triangle begins:
==============================================
...... k: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10 11 12
==============================================
n=1 ..... 1,
n=2 ..... 2, 1,
n=3 ..... 3, 0, 1,
n=4 ..... 5, 2, 0, 1,
n=5 ..... 7, 0, 0, 0, 1,
n=6 .... 11, 3, 2, 0, 0, 1,
n=7 .... 15, 0, 0, 0, 0, 0, 1,
n=8 .... 22, 5, 0, 2, 0, 0, 0, 1,
n=9 .... 30, 0, 3, 0, 0, 0, 0, 0, 1,
n=10 ... 42, 7, 0, 0, 2, 0, 0, 0, 0, 1,
n=11 ... 56, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
n=12 ... 77,11, 5, 3, 0, 2, 0, 0, 0, 0, 0, 1,
...
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MATHEMATICA
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T[n_, k_]:= If[IntegerQ[n/k], PartitionsP[n/k], 0];
Table[T[n, k], {n, 15}, {k, n}]//Flatten (* G. C. Greubel, Jan 12 2023 *)
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PROG
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(SageMath)
def A168021(n, k): return number_of_partitions(n/k) if (n%k)==0 else 0
flatten([[A168021(n, k) for k in range(1, n+1)] for n in range(1, 16)]) # G. C. Greubel, Jan 12 2023
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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