|
|
A344824
|
|
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).
|
|
7
|
|
|
1, 1, 2, 1, 1, 3, 1, 0, 3, 4, 1, -1, 5, 2, 5, 1, -2, 9, -4, 4, 6, 1, -3, 15, -20, 13, 4, 7, 1, -4, 23, -52, 62, -16, 6, 8, 1, -5, 33, -106, 205, -174, 49, 4, 9, 1, -6, 45, -188, 520, -806, 556, -88, 7, 10, 1, -7, 59, -304, 1109, -2584, 3291, -1660, 173, 7, 11
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,3
|
|
LINKS
|
|
|
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) * (k-n+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, 1, 0, -1, -2, -3, -4, ...
3, 3, 5, 9, 15, 23, 33, ...
4, 2, -4, -20, -52, -106, -188, ...
5, 4, 13, 62, 205, 520, 1109, ...
6, 4, -16, -174, -806, -2584, -6636, ...
Antidiagonal triangle, T(n, k), begins:
1;
1, 2;
1, 1, 3;
1, 0, 3, 4;
1, -1, 5, 2, 5;
1, -2, 9, -4, 4, 6;
1, -3, 15, -20, 13, 4, 7;
1, -4, 23, -52, 62, -16, 6, 8;
1, -5, 33, -106, 205, -174, 49, 4, 9;
1, -6, 45, -188, 520, -806, 556, -88, 7, 10;
|
|
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]]) >;
A344824:= func< n, k | A(k+1, n-k-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 A344824(n, k): return A(k+1, n-k-1)
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|