OFFSET
1,4
LINKS
G. C. Greubel, Rows n = 1..100 of the triangle, flattened
FORMULA
Sum_{k=1..n} T(n, k) = A000027(n) (row sums).
T(n, 1) = A059851(n).
From G. C. Greubel, Jun 23 2024: (Start)
T(n, k) = A010766(n,k) * AA130595(n-1, k-1) as infinite lower triangular matrices.
T(n, k) = Sum_{j=k..n} (-1)^(j+k) * floor(n/j) * binomial(j-1, k-1).
T(2*n-1, n) = (-1)^(n-1)*A026641(n).
T(2*n-2, n-1) = (-1)^n*A014300(n-1), for n >= 2.
Sum_{k=1..n} (-1)^(k-1)*T(n, k) = A344817(n).
Sum_{k=1..n} k*T(n, k) = A032766(n-1).
Sum_{k=1..n} (k+1)*T(n, k) = A047215(n). (End)
EXAMPLE
First few rows of the triangle:
1;
1, 1;
3, -1, 1;
2, 3 -2, 1;
4, -1, 4, -3, 1;
4, 3, -5, 7, -4, 1;
6, -3, 10, -13, 11, -5, 1;
4, 8, -14, 23, -24, 16, -6, 1;
...
MATHEMATICA
T[n_, k_]:= Sum[(-1)^(j+k)*Floor[n/j]*Binomial[j-1, k-1], {j, k, n}];
Table[T[n, k], {n, 15}, {k, n}]//Flatten (* G. C. Greubel, Jun 23 2024 *)
PROG
(Magma)
A128316:= func< n, k | (&+[(-1)^(j+k)*Floor(n/j)*Binomial(j-1, k-1): j in [k..n]]) >;
[A128316(n, k): k in [1..n], n in [1..15]]; // G. C. Greubel, Jun 23 2024
(SageMath)
def A128316(n, k): return sum((-1)^(j+k)*int(n//j)*binomial(j-1, k-1) for j in range(k, n+1))
flatten([[A128316(n, k) for k in range(1, n+1)] for n in range(1, 16)]) # G. C. Greubel, Jun 23 2024
CROSSREFS
KEYWORD
tabl,sign
AUTHOR
Gary W. Adamson, Feb 25 2007
EXTENSIONS
a(28) = 1 inserted and more terms from Georg Fischer, Jun 06 2023
STATUS
approved