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A168016
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Triangle T(n,k) read by rows in which row n list the number of partitions of n into parts divisible by k for k=n,n-1,...,1.
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8
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1, 1, 2, 1, 0, 3, 1, 0, 2, 5, 1, 0, 0, 0, 7, 1, 0, 0, 2, 3, 11, 1, 0, 0, 0, 0, 0, 15, 1, 0, 0, 0, 2, 0, 5, 22, 1, 0, 0, 0, 0, 0, 3, 0, 30, 1, 0, 0, 0, 0, 2, 0, 0, 7, 42, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 56, 1, 0, 0, 0, 0, 0, 2, 0, 3, 5, 11, 77, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 101
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OFFSET
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1,3
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LINKS
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FORMULA
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T(n, k) = A000041(n/k) if k|n; otherwise T(n,k) = 0.
Sum_{k=2..n-1} T(n, k) = A168111(n-1). (End)
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EXAMPLE
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Triangle begins:
==============================================
.... k: 12 11 10. 9. 8. 7. 6. 5. 4. 3.. 2.. 1.
==============================================
n=1 ....................................... 1,
n=2 ................................... 1, 2,
n=3 ............................... 1, 0, 3,
n=4 ............................ 1, 0, 2, 5,
n=5 ......................... 1, 0, 0, 0, 7,
n=6 ...................... 1, 0, 0, 2, 3, 11,
n=7 ................... 1, 0, 0, 0, 0, 0, 15,
n=8 ................ 1, 0, 0, 0, 2, 0, 5, 22,
n=9 ............. 1, 0, 0, 0, 0, 0, 3, 0, 30,
n=10 ......... 1, 0, 0, 0, 0, 2, 0, 0, 7, 42,
n=11 ...... 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 56,
n=12 ... 1, 0, 0, 0, 0, 0, 2, 0, 3, 5, 11, 77,
...
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MATHEMATICA
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T[n_, k_]:= If[IntegerQ[n/(n-k+1)], PartitionsP[n/(n-k+1)], 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 T(n, k): return number_of_partitions(n/(n-k+1)) if (n%(n-k+1))==0 else 0
flatten([[T(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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