A177229 Triangle, read by rows, T(n, k) = -binomial(n,k) for 1 <= k <= n-1, otherwise T(n, k) = 4.

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

Offset: 0

Author

Roger L. Bagula, May 05 2010

Keywords

• sign
• tabl
• less
• easy

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 (A177227), t = 1/3 (A177228), and t = 1/4 (this sequence).

Examples

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

Links

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

Crossrefs

Cf. A007318, A177227, A177228.

Programs

Magma

A177229:= func< n,k | k eq 0 or k eq n select 4 else -Binomial(n,k) >;
[A177229(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 09 2024

Mathematica

T[n_, k_]:= If[k==0 || k==n, 4, -Binomial[n,k]];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten

SageMath

def A177229(n,k): return 4 if (k==0 or k==n) else -binomial(n,k)
flatten([[A177229(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Apr 09 2024

Formula

T(n, k) = -binomial(n,k) for 1 <= k <= n-1, otherwise T(n, k) = 4.

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

Sum_{k=0..n} T(n, k) = 10 - 2^n - 5*[n=0] (row sums).

Sum_{k=0..n} (-1)^k*T(n, k) = 5*(1 + (-1)^n) - 6*[n=0].

Sum_{k=0..floor(n/2)} T(n-k,k) = (5/2)*(3+(-1)^n-2*[n=0])-Fibonacci(n+1).

Sum_{k=0..floor(n/2)} (-1)^k*T(n-k,k) = 5*(1 + cos(n*Pi/2) - [n=0]) - (2/sqrt(3))*cos((2*n-1)*Pi/6). (End)

Extensions

Edited by G. C. Greubel, Apr 09 2024