A177227 Triangle, read by rows, T(n, k) = -binomial(n,k) for 0 < k < n, otherwise T(n, k) = 2.

2, 2, 2, 2, -2, 2, 2, -3, -3, 2, 2, -4, -6, -4, 2, 2, -5, -10, -10, -5, 2, 2, -6, -15, -20, -15, -6, 2, 2, -7, -21, -35, -35, -21, -7, 2, 2, -8, -28, -56, -70, -56, -28, -8, 2, 2, -9, -36, -84, -126, -126, -84, -36, -9, 2, 2, -10, -45, -120, -210, -252, -210, -120, -45, -10, 2

Offset

0,1

Comments

This triangle may also be constructed in the following way. Let f_{n}(t) = d^n/dt^n (t/(1+t) = (-1)^(n+1)*n!*(1+t)^(-n-1). Then the triangle is given as f_{n}(t)/((1+t)*f_{k}(t)*f_{n-k}(t)) when t = 1/2 (this sequence), t = 1/3 (A177228), and t = 1/4 (A177229).

Links

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

Formula

T(n, 0) = T(n, n) = 2, otherwise T(n, k) = -binomial(n,k).

Sum_{k=0..n} T(n, k) = -A131130(n-2) - 3*[n=0], n >= 1 (row sums).

From G. C. Greubel, Apr 09 2024: (Start)

Sum_{k=0..n} (-1)^k*T(n, k) = 3*(1 + (-1)^n) - 4*[n=0].

Sum_{k=0..floor(n/2)} T(n-k,k) = (3/2)*(3 + (-1)^n - 2*[n=0])-Fibonacci(n+1).

Sum_{k=0..floor(n/2)} (-1)^k*T(n-k,k) = 3*(1 + cos(n*Pi/2) - [n=0]) - (2/sqrt(3))*cos((2*n-1)*Pi/6). (End)

Example

Triangle begins as:
  2;
  2,   2;
  2,  -2,   2;
  2,  -3,  -3,    2;
  2,  -4,  -6,   -4,    2;
  2,  -5, -10,  -10,   -5,    2;
  2,  -6, -15,  -20,  -15,   -6,    2;
  2,  -7, -21,  -35,  -35,  -21,   -7,    2;
  2,  -8, -28,  -56,  -70,  -56,  -28,   -8,   2;
  2,  -9, -36,  -84, -126, -126,  -84,  -36,  -9,   2;
  2, -10, -45, -120, -210, -252, -210, -120, -45, -10,  2;

Mathematica

T[n_, k_]:= If[k==0 || k==n, 2, -Binomial[n, k]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten

Prog

(Magma)
A177227:= func< n, k | k eq 0 or k eq n select 2 else -Binomial(n, k) >;
[A177227(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 09 2024
(SageMath)
def A177227(n, k): return 2 if (k==0 or k==n) else -binomial(n, k)
flatten([[A177227(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Apr 09 2024

Crossrefs

Cf. A007318, A131130 (related to row sums), A177228, A177229.

Sequence in context: A306240 A109829 A054125 * A174373 A232270 A191517

Adjacent sequences: A177224 A177225 A177226 * A177228 A177229 A177230

Keyword

sign,tabl,less,easy,changed

Author

Roger L. Bagula, May 05 2010

Extensions

Edited by G. C. Greubel, Apr 09 2024

Status

approved