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A308813
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Square array A(n,k), n >= 1, k >= 0, read by descending antidiagonals, where A(n,k) is Sum_{d|n} k^(d-1).
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8
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1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 4, 5, 3, 1, 1, 5, 10, 11, 2, 1, 1, 6, 17, 31, 17, 4, 1, 1, 7, 26, 69, 82, 39, 2, 1, 1, 8, 37, 131, 257, 256, 65, 4, 1, 1, 9, 50, 223, 626, 1045, 730, 139, 3, 1, 1, 10, 65, 351, 1297, 3156, 4097, 2218, 261, 4, 1
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OFFSET
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1,5
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LINKS
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FORMULA
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G.f. of column k: Sum_{j>=1} x^j/(1 - k*x^j).
T(n, k) = Sum_{d|(k+1)} (n-k-1)^(d-1), with T(n, n) = 1. - G. C. Greubel, Jun 26 2024
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EXAMPLE
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Square array, A(n,k), begins:
1, 1, 1, 1, 1, 1, 1, ...
1, 2, 3, 4, 5, 6, 7, ...
1, 2, 5, 10, 17, 26, 37, ...
1, 3, 11, 31, 69, 131, 223, ...
1, 2, 17, 82, 257, 626, 1297, ...
1, 4, 39, 256, 1045, 3156, 7819, ...
1, 2, 65, 730, 4097, 15626, 46657, ...
Antidiagonal triangle, T(n,k), begins as:
1;
1, 1;
1, 2, 1;
1, 3, 2, 1;
1, 4, 5, 3, 1;
1, 5, 10, 11, 2, 1;
1, 6, 17, 31, 17, 4, 1;
1, 7, 26, 69, 82, 39, 2, 1;
1, 8, 37, 131, 257, 256, 65, 4, 1;
1, 9, 50, 223, 626, 1045, 730, 139, 3, 1;
1, 10, 65, 351, 1297, 3156, 4097, 2218, 261, 4, 1;
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MATHEMATICA
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A[n_, k_] := DivisorSum[n, If[k == # - 1 == 0, 1, k^(# - 1)] &];
Table[A[k + 1, n - k - 1], {n, 1, 11}, {k, 0, n - 1}] // Flatten (* Amiram Eldar, May 07 2021 *)
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PROG
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(Magma)
A:= func< n, k | (&+[k^(d-1): d in Divisors(n)]) >;
A308813:= func< n, k | A(k+1, n-k-1) >;
(SageMath)
def A(n, k): return sum(k^(j-1) for j in (1..n) if (j).divides(n))
def A308813(n, k): return A(k+1, n-k-1)
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CROSSREFS
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Columns k=0..10 give A000012, A000005, A034729, A034730, A339684, A339685, A339686, A339687, A339688, A339689, A113999.
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KEYWORD
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AUTHOR
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STATUS
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approved
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