A344821 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals downwards, where A(n,k) = Sum_{j=1..n} floor(n/j) * k^(j-1).
1, 1, 2, 1, 3, 3, 1, 4, 5, 4, 1, 5, 9, 8, 5, 1, 6, 15, 20, 10, 6, 1, 7, 23, 46, 37, 14, 7, 1, 8, 33, 92, 128, 76, 16, 8, 1, 9, 45, 164, 349, 384, 141, 20, 9, 1, 10, 59, 268, 790, 1394, 1114, 280, 23, 10, 1, 11, 75, 410, 1565, 3946, 5491, 3332, 541, 27, 11
Offset: 1
Examples
Square array, A(n, k), begins: 1, 1, 1, 1, 1, 1, 1, ... 2, 3, 4, 5, 6, 7, 8, ... 3, 5, 9, 15, 23, 33, 45, ... 4, 8, 20, 46, 92, 164, 268, ... 5, 10, 37, 128, 349, 790, 1565, ... 6, 14, 76, 384, 1394, 3946, 9384, ... Antidiagonal triangle, T(n, k), begins: 1; 1, 2; 1, 3, 3; 1, 4, 5, 4; 1, 5, 9, 8, 5; 1, 6, 15, 20, 10, 6; 1, 7, 23, 46, 37, 14, 7; 1, 8, 33, 92, 128, 76, 16, 8; 1, 9, 45, 164, 349, 384, 141, 20, 9;
Links
- G. C. Greubel, Antidiagonals n = 1..50, flattened
Crossrefs
Programs
-
Magma
A:= func< n,k | k eq n select n else (&+[Floor(n/j)*k^(j-1): j in [1..n]]) >; A344821:= func< n,k | A(k+1, n-k-1) >; [A344821(n,k): k in [0..n-1], n in [1..12]]; // G. C. Greubel, Jun 27 2024
-
Mathematica
A[n_, k_] := Sum[If[k == 0 && j == 1, 1, k^(j - 1)] * Quotient[n, j], {j, 1, n}]; Table[A[k, n - k], {n, 1, 10}, {k, 1, n}] // Flatten (* Amiram Eldar, May 29 2021 *) -
PARI
A(n, k) = sum(j=1, n, n\j*k^(j-1));
-
PARI
A(n, k) = sum(j=1, n, sumdiv(j, d, k^(d-1)));
-
SageMath
def A(n,k): return n if k==n else sum((n//j)*k^(j-1) for j in range(1,n+1)) def A344821(n,k): return A(k+1, n-k-1) flatten([[A344821(n,k) for k in range(n)] for n in range(1,13)]) # G. C. Greubel, Jun 27 2024
Formula
G.f. of column k: (1/(1 - x)) * Sum_{j>=1} x^j/(1 - k*x^j).
G.f. of column k: (1/(1 - x)) * Sum_{j>=1} k^(j-1) * x^j/(1 - x^j).
A(n, k) = Sum_{j=1..n} Sum_{d|j} k^(d - 1).
T(n, k) = Sum_{j=1..k+1} floor((k+1)/j) * (n-k-1)^(j-1), for n >= 1, 0 <= k <= n-1 (antidiagonal triangle). - G. C. Greubel, Jun 27 2024