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reviewed
approved
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Revision History for A308813
(Bold, blue-underlined text is an addition; faded, red-underlined text is aShowing entries 1-10 | older changes
Square array A(n,k), n >= 1, k >= 0, read by descending antidiagonals, where A(n,k) is Sum_{d|n} k^(d-1).
(history; published version)
(history; published version)
STATUS
proposed
reviewed
STATUS
editing
proposed
NAME
Square array A(n,k), n >= 1, k >= 0, read by descending antidiagonals, where A(n,k) is Sum_{d|n} k^(d-1).
STATUS
proposed
editing
STATUS
editing
proposed
FORMULA
T(n, k) = Sum_{d|(k+1)} (n-k-1)^(d-1), with T(n, n) = 1. - G. C. Greubel, Jun 26 2024
EXAMPLE
Square array , A(n,k), begins:
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;
MATHEMATICA
TA[n_, k_] := DivisorSum[n, If[k == # - 1 == 0, 1, k^(# - 1)] &]; Table[T[k + 1, n - k - 1], {n, 1, 11}, {k, 0, n - 1}] // Flatten (* _Amiram Eldar_, May 07 2021 *)
Table[A[k + 1, n - k - 1], {n, 1, 11}, {k, 0, n - 1}] // Flatten (* Amiram Eldar, May 07 2021 *)
PROG
(Magma)
A:= func< n, k | (&+[k^(d-1): d in Divisors(n)]) >;
A308813:= func< n, k | A(k+1, n-k-1) >;
[A308813(n, k): k in [0..n-1], n in [1..12]]; // G. C. Greubel, Jun 26 2024
(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)
flatten([[A308813(n, k) for k in range(n)] for n in range(1, 13)]) # G. C. Greubel, Jun 26 2024
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approved
editing
STATUS
reviewed
approved
STATUS
proposed
reviewed
STATUS
editing
proposed
MATHEMATICA
T[n_, k_] := DivisorSum[n, If[k == # - 1 == 0, 1, k^(# - 1)] &]; Table[T[k + 1, n - k - 1], {n, 1, 11}, {k, 0, n - 1}] // Flatten (* Amiram Eldar, May 07 2021 *)
STATUS
approved
editing