A061177 Coefficients of polynomials ( (1 -x +sqrt(x))^(n+1) - (1 -x -sqrt(x))^(n+1) )/(2*sqrt(x)).

1, 2, -2, 3, -5, 3, 4, -8, 8, -4, 5, -10, 11, -10, 5, 6, -10, 6, -6, 10, -6, 7, -7, -14, 29, -14, -7, 7, 8, 0, -56, 120, -120, 56, 0, -8, 9, 12, -126, 288, -365, 288, -126, 12, 9, 10, 30, -228, 540, -770, 770, -540, 228, -30, -10, 11, 55, -363, 858

Offset: 0

Author

Wolfdieter Lang, Apr 20 2001

Keywords

• sign
• easy
• tabl

Comments

The row polynomial pFo(m,x) = Sum_{j=0..m} T(m, j)*x^j is the numerator of the g.f. for the m-th column sequence of A060921, the odd part of the bisected Fibonacci triangle.

Examples

			The first few polynomials are:
pFo(0, x) = 1.
pFo(1, x) = 2 -  2*x.
pFo(2, x) = 3 -  5*x +  3*x^2.
pFo(3, x) = 4 -  8*x +  8*x^2 -  4*x^3.
pFo(4, x) = 5 - 10*x + 11*x^2 - 10*x^3 +  5*x^4.
pFo(5, x) = 6 - 10*x +  6*x^2 -  6*x^3 + 10*x^4 - 6*x^5.
Number triangle begins as:
   1;
   2,  -2;
   3,  -5,    3;
   4,  -8,    8,  -4;
   5, -10,   11, -10,    5;
   6, -10,    6,  -6,   10,  -6;
   7,  -7,  -14,  29,  -14,  -7,    7;
   8,   0,  -56, 120, -120,  56,    0,  -8;
   9,  12, -126, 288, -365, 288, -126,  12,   9;
  10,  30, -228, 540, -770, 770, -540, 228, -30, -10;
		

Links

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

Crossrefs

Cf. A059841, A060921, A061176 (companion triangle).

Programs

Magma

A061177:= func< n,k | (&+[(-1)^(k+j)*Binomial(n+1,2*j+1)*Binomial(n-2*j,k-j): j in [0..k]]) >;
[A061177(n,k): k in [0..n], n in [0..15]]; // G. C. Greubel, Apr 06 2021

Mathematica

T[n_, k_]:= Sum[(-1)^(k-j)*Binomial[n+1, 2*j+1]*Binomial[n-2*j, k-j], {j,0,k}];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Apr 06 2021 *)

Sage

def A061177(n,k): return sum((-1)^(k+j)*binomial(n+1,2*j+1)*binomial(n-2*j,k-j) for j in (0..k))
flatten([[A061177(n,k) for k in (0..n)] for n in (0..15)]) # G. C. Greubel, Apr 06 2021

Formula

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

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

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