A308813 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals downwards, where A(n,k) is Sum_{d|n} k^(d-1).
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
Offset: 1
Examples
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;
Links
- Seiichi Manyama, Antidiagonals n = 1..140, flattened
Crossrefs
Programs
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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) >; [A308813(n,k): k in [0..n-1], n in [1..12]]; // G. C. Greubel, Jun 26 2024
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Mathematica
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 *) -
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
Formula
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