A128316 Triangle read by rows: A000012 * A128315 as infinite lower triangular matrices.
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, 7, -2, 15, -33, 46, -40, 22, -7, 1, 7, 4, -15, 47, -79, 86, -62, 29, -8, 1, 9, -6, 30, -73, 131, -166, 148, -91, 37, -9, 1, 7, 12, -37, 103, -204, 297, -314, 239, -128, 46, -10, 1
Offset: 1
Examples
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; ...
Links
- G. C. Greubel, Rows n = 1..100 of the triangle, flattened
Crossrefs
Programs
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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
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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 *) -
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
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) = 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)
Extensions
a(28) = 1 inserted and more terms from Georg Fischer, Jun 06 2023
Comments