A108086 Triangle, read by rows, where T(0,0) = 1, T(n,k) = (-1)^(n+k)*T(n-1,k) + T(n-1,k-1); a signed version of Pascal's triangle.
1, -1, 1, -1, -2, 1, 1, -3, -3, 1, 1, 4, -6, -4, 1, -1, 5, 10, -10, -5, 1, -1, -6, 15, 20, -15, -6, 1, 1, -7, -21, 35, 35, -21, -7, 1, 1, 8, -28, -56, 70, 56, -28, -8, 1, -1, 9, 36, -84, -126, 126, 84, -36, -9, 1, -1, -10, 45, 120, -210, -252, 210, 120, -45, -10, 1, 1, -11, -55, 165, 330, -462, -462, 330, 165, -55, -11, 1
Offset: 0
Links
- G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Programs
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Magma
A108086:= func< n,k | (-1)^Floor((n-k+1)/2)*Binomial(n,k) >; [A108086(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 02 2022
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Mathematica
A108086[n_, k_]:= (-1)^(Floor[(n-k+1)/2])*Binomial[n, k]; Table[A108086[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Dec 02 2022 *)
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SageMath
def A108086(n,k): return (-1)^int((n-k+1)/2)*binomial(n,k) flatten([[A108086(n,k) for k in range(n+1)] for n in range(14)]) # G. C. Greubel, Dec 02 2022
Formula
T(n,k) = (-1)^(n+k)*T(n-1,k) + T(n-1,k-1), with T(0, 0) = 1.
T(n, k) = (-1)^floor((n-k+1)/2) * A007318(n, k).
From G. C. Greubel, Dec 02 2022: (Start)
T(2*n, n) = (-1)^binomial(n+1,2) * A000984(n).
T(2*n, n+1) = (-1)^binomial(n,2) * A001791(n), n >= 1.
T(2*n, n-1) = (-1)^binomial(n+2,2) * A001791(n).
T(2*n+1, n-1) = (-1)^binomial(n-1,2) * A002054(n).
T(2*n+1, n+1) = (-1)^binomial(n+1,2) * A001700(n+1).
Sum_{k=0..n} T(n, k) = (-1)^n * A090132(n).
Sum_{k=0..n} (-1)^k * T(n, k) = A108520(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = (-1)^n * A260192(n-1).
Sum_{k=0..floor(n/2)} (-1)^k * T(n-k, k) = A333378(n+1). (End)