A320508 T(n,k) = binomial(n - k - 1, k), 0 <= k < n, and T(n,n) = (-1)^n, triangle read by rows.

1, 1, -1, 1, 0, 1, 1, 1, 0, -1, 1, 2, 0, 0, 1, 1, 3, 1, 0, 0, -1, 1, 4, 3, 0, 0, 0, 1, 1, 5, 6, 1, 0, 0, 0, -1, 1, 6, 10, 4, 0, 0, 0, 0, 1, 1, 7, 15, 10, 1, 0, 0, 0, 0, -1, 1, 8, 21, 20, 5, 0, 0, 0, 0, 0, 1, 1, 9, 28, 35, 15, 1, 0, 0, 0, 0, 0, -1, 1, 10, 36

Offset: 0

Author

Franck Maminirina Ramaharo, Oct 14 2018

Keywords

• sign
• easy
• tabl

Comments

Differs from A164925 in signs.

The n-th row consists of the coefficients in the expansion of (-x)^n + (((1 + sqrt(1 + 4*x))/2)^n -((1 - sqrt(1 + 4*x))/2)^n )/sqrt(1 + 4*x).

The coefficients in the expansion of Sum_{j=0..floor((n - 1)/2)} T(n,k)*x^(n - 2*j - 1) yield the n-th row in A168561, the coefficients of the n-th Fibonacci polynomial.

Row n sums up to Fibonacci(n) + (-1)^n (A008346).

Examples

			Triangle begins:
    1;
    1, -1;
    1,  0,  1;
    1,  1,  0, -1;
    1,  2,  0,  0,  1;
    1,  3,  1,  0,  0, -1;
    1,  4,  3,  0,  0,  0, 1;
    1,  5,  6,  1,  0,  0, 0, -1;
    1,  6, 10,  4,  0,  0, 0,  0, 1;
    1,  7, 15, 10,  1,  0, 0,  0, 0, -1;
    1,  8, 21, 20,  5,  0, 0,  0, 0,  0, 1;
    1,  9, 28, 35, 15,  1, 0,  0, 0,  0, 0, -1;
    ...
		

Links

• Wikipedia, Fibonacci polynomials

Crossrefs

Inspired by A123018.

Cf. A007318, A026729, A049310, A052553, A164925, A168561.

Programs

Mathematica

Table[Table[Binomial[n - k - 1, k], {k, 0, n}], {n, 0, 12}]//Flatten

Maxima

create_list(binomial(n - k - 1, k), n, 0, 12, k, 0, n);

Formula

G.f.: 1/((1 + x*y)*(1 - y - x*y^2)).

E.g.f.: exp(-x*y) + (exp(y*(1 + sqrt(1 + 4*x))/2) - exp(y*(1 - sqrt(1 + 4*x))/2))/sqrt(1 + 4*x).

T(n,1) = A023443(n).