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

0,5

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

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)

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 *)

Prog

(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
(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

Crossrefs

Cf. A000984, A001700, A002054, A007318, A009116.

Cf. A090132, A108520, A260192, A333378.

Sequence in context: A117440 A118433 A007318 * A130595 A108363 A329052

Adjacent sequences: A108083 A108084 A108085 * A108087 A108088 A108089

Keyword

sign,tabl

Author

Gerald McGarvey, Jun 05 2005

Status

approved