A061176 Coefficients of polynomials ( (1 -x +sqrt(x))^n + (1 -x -sqrt(x))^n )/2.

1, 1, -1, 1, -1, 1, 1, 0, 0, -1, 1, 2, -5, 2, 1, 1, 5, -15, 15, -5, -1, 1, 9, -30, 41, -30, 9, 1, 1, 14, -49, 77, -77, 49, -14, -1, 1, 20, -70, 112, -125, 112, -70, 20, 1, 1, 27, -90, 126, -117, 117, -126, 90, -27, -1, 1, 35, -105, 90, 45, -131, 45, 90, -105, 35, 1

Offset

0,12

Comments

The row polynomial pFe(k+1, x) = Sum_{j=0..k+1} T(k+1, j)*x^j is the numerator of the g.f. for the k-th column sequence of A060920, the even part of the bisected Fibonacci triangle.

Links

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

Formula

T(n, k) = coefficients of x^k of ((1-x+sqrt(x))^n + (1-x-sqrt(x))^n)/2.

T(n, k) = Sum_{j=0..k} (-1)^(k-j)*binomial(n, 2*j)*binomial(n-2*j, k-j), if 0 <= k <= floor(n/2) and T(n, k) = (-1)^n*T(n, n-k) if floor(n/2) < k <= n, otherwise 0.

Sum_{k=0..n} T(n, k) = A059841(n) = (1 + (-1)^n)/2. - G. C. Greubel, Apr 06 2021

Example

The first few polynomials are:
pFe(0,x) = 1.
pFe(1,x) = 1 -   x.
pFe(2,x) = 1 -   x +   x^2.
pFe(3,x) = 1 - 0*x + 0*x^2 -   x^3.
pFe(4,x) = 1 + 2*x - 5*x^2 + 2*x^3 + x^4.
Number triangle begins as:
  1;
  1, -1;
  1, -1,   1;
  1,  0,   0,  -1;
  1,  2,  -5,   2,    1;
  1,  5, -15,  15,   -5,  -1;
  1,  9, -30,  41,  -30,   9,   1;
  1, 14, -49,  77,  -77,  49, -14, -1;
  1, 20, -70, 112, -125, 112, -70, 20, 1;

Mathematica

T[n_, k_]:= Sum[(-1)^(k+j)*Binomial[n, 2*j]*Binomial[n-2*j, k-j], {j, 0, k}];
Table[T[n, k], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Apr 06 2021 *)

Prog

(Magma)
A061176:= func< n, k | (&+[(-1)^(k+j)*Binomial(n, 2*j)*Binomial(n-2*j, k-j): j in [0..k]]) >;
[A061176(n, k): k in [0..n], n in [0..15]]; // G. C. Greubel, Apr 06 2021
(Sage)
def A061176(n, k): return sum((-1)^(k+j)*binomial(n, 2*j)*binomial(n-2*j, k-j) for j in (0..k))
flatten([[A061176(n, k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Apr 06 2021

Crossrefs

Cf. A059841, A060920, A061177 (companion triangle), A180957.

Sequence in context: A320032 A270061 A342059 * A180957 A124780 A369872

Adjacent sequences: A061173 A061174 A061175 * A061177 A061178 A061179

Keyword

sign,easy,tabl

Author

Wolfdieter Lang, Apr 20 2001

Status

approved