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

1,3

Links

G. C. Greubel, Antidiagonals n = 1..50, flattened

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

Example

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;

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 *)

Prog

(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)));
(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
(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

Crossrefs

Columns k=0..5 give A000027, A006218, A268235, A344814, A344815, A344816.

A(n,n) gives A332533.

Cf. A308813, A344824.

Sequence in context: A132108 A377000 A210489 * A125175 A210552 A193376

Adjacent sequences: A344818 A344819 A344820 * A344822 A344823 A344824

Keyword

nonn,tabl

Author

Seiichi Manyama, May 29 2021

Status

approved