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
KEYWORD
AUTHOR
Roger L. Bagula, May 05 2010
EXTENSIONS
Edited by G. C. Greubel, Apr 09 2024
STATUS
approved