A177228 Triangle read by rows: T(n, k) = -binomial(n,k) for 1 <= k <= n-1, otherwise T(n, k) = 3.

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

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 (A177227), t = 1/3 (this sequence), and t = 1/4 (A177229).

This is the Pascal triangle A007318, with all entries sign-flipped, and 3's inserted at the beginning and end of each row. - R. J. Mathar, Mar 27 2024

Links

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

Formula

T(n, k) = -binomial(n,k) for 1 <= k <= n-1, otherwise T(n, k) = 3.

Sum_{k=0..n} T(n, k) = 8 - 2^n, for n >= 1.

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

Sum_{k=0..n} (-1)^k*T(n, k) = 4*(1 + (-1)^n) - 5*[n=0].

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

Sum_{k=0..floor(n/2)} (-1)^k*T(n-k,k) = 4*(1 + cos(n*Pi/2) - [n=0]) - (2/sqrt(3))*cos((2*n-1)*Pi/6). (End)

Example

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

Maple

f := proc(n, t)
    if n = 0 then
        t/(1+t) ;
    else
        diff( t/(1+t), t$n) ;
        factor(%) ;
    end if;
end proc:
A177228 := proc(n, m)
    f(n, t)/f(m, t)/f(n-m, t) ;
    %/(1+t) ;
    subs(t=1/3, %) ;
end proc:
seq(seq( A177228(n, m), m=0..n), n=0..12) ; # R. J. Mathar, Mar 27 2024

Mathematica

T[n_, k_]:= If[k==0 || k==n, 3, -Binomial[n, k]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten

Prog

(Magma)
A177228:= func< n, k | k eq 0 or k eq n select 3 else -Binomial(n, k) >;
[A177228(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Apr 09 2024
(SageMath)
def A177228(n, k): return 3 if (k==0 or k==n) else -binomial(n, k)
flatten([[A177228(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Apr 09 2024

Crossrefs

Cf. A007318, A177227, A177229.

Sequence in context: A183051 A031355 A337265 * A247655 A097675 A141605

Adjacent sequences: A177225 A177226 A177227 * A177229 A177230 A177231

Keyword

sign,tabl,less,easy

Author

Roger L. Bagula, May 05 2010

Extensions

Edited by G. C. Greubel, Apr 09 2024

Status

approved