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A339106
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Triangle read by rows: T(n,k) = A000203(n-k+1)*A000041(k-1), n >= 1, 1 <= k <= n.
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17
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1, 3, 1, 4, 3, 2, 7, 4, 6, 3, 6, 7, 8, 9, 5, 12, 6, 14, 12, 15, 7, 8, 12, 12, 21, 20, 21, 11, 15, 8, 24, 18, 35, 28, 33, 15, 13, 15, 16, 36, 30, 49, 44, 45, 22, 18, 13, 30, 24, 60, 42, 77, 60, 66, 30, 12, 18, 26, 45, 40, 84, 66, 105, 88, 90, 42, 28, 12, 36, 39, 75, 56, 132, 90, 154, 120, 126, 56
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OFFSET
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1,2
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COMMENTS
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Conjecture 1: T(n,k) is the sum of all divisors of all (n - k + 1)'s in the n-th row of triangle A176206, assuming that A176206 has offset 1. The same for the triangle A340061.
Conjecture 2: the sum of row n equals A066186(n), the sum of all parts of all partitions of n.
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LINKS
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FORMULA
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T(n,k) = sigma(n-k+1)*p(k-1), n >= 1, 1 <= k <= n.
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EXAMPLE
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Triangle begins:
1;
3, 1;
4, 3, 2;
7, 4, 6, 3;
6, 7, 8, 9, 5;
12, 6, 14, 12, 15, 7;
8, 12, 12, 21, 20, 21, 11;
15, 8, 24, 18, 35, 28, 33, 15;
13, 15, 16, 36, 30, 49, 44, 45, 22;
18, 13, 30, 24, 60, 42, 77, 60, 66, 30;
12, 18, 26, 45, 40, 84, 66, 105, 88, 90, 42;
28, 12, 36, 39, 75, 56, 132, 90, 154, 120, 126, 56;
...
For n = 6 the calculation of every term of row 6 is as follows:
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1 1 * 12 = 12
2 1 * 6 = 6
3 2 * 7 = 14
4 3 * 4 = 12
5 5 * 3 = 15
6 7 * 1 = 7
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The sum of row 6 is 12 + 6 + 14 + 12 + 15 + 7 = 66, equaling A066186(6).
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MATHEMATICA
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T[n_, k_] := DivisorSigma[1, n - k + 1] * PartitionsP[k - 1]; Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Amiram Eldar, Jan 08 2021 *)
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PROG
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(PARI) T(n, k) = sigma(n-k+1)*numbpart(k-1); \\ Michel Marcus, Jan 08 2021
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CROSSREFS
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Row sums give A066186 (conjectured).
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KEYWORD
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AUTHOR
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STATUS
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approved
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