A061176 - OEIS

%I #10 Apr 06 2021 23:09:26

%S 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,

%T 1,1,14,-49,77,-77,49,-14,-1,1,20,-70,112,-125,112,-70,20,1,1,27,-90,

%U 126,-117,117,-126,90,-27,-1,1,35,-105,90,45,-131,45,90,-105,35,1

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

%C 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.

%H G. C. Greubel, <a href="/A061176/b061176.txt">Rows n = 0..50 of the triangle, flattened</a>

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

%F 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.

%F Sum_{k=0..n} T(n, k) = A059841(n) = (1 + (-1)^n)/2. - _G. C. Greubel_, Apr 06 2021

%e The first few polynomials are:

%e pFe(0,x) = 1.

%e pFe(1,x) = 1 -   x.

%e pFe(2,x) = 1 -   x +   x^2.

%e pFe(3,x) = 1 - 0*x + 0*x^2 -   x^3.

%e pFe(4,x) = 1 + 2*x - 5*x^2 + 2*x^3 + x^4.

%e Number triangle begins as:

%e   1;

%e   1, -1;

%e   1, -1,   1;

%e   1,  0,   0,  -1;

%e   1,  2,  -5,   2,    1;

%e   1,  5, -15,  15,   -5,  -1;

%e   1,  9, -30,  41,  -30,   9,   1;

%e   1, 14, -49,  77,  -77,  49, -14, -1;

%e   1, 20, -70, 112, -125, 112, -70, 20, 1;

%t T[n_, k_]:= Sum[(-1)^(k+j)*Binomial[n, 2*j]*Binomial[n-2*j, k-j], {j,0,k}];

%t Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* _G. C. Greubel_, Apr 06 2021 *)

%o (Magma)

%o A061176:= func< n,k | (&+[(-1)^(k+j)*Binomial(n,2*j)*Binomial(n-2*j,k-j): j in [0..k]]) >;

%o [A061176(n,k): k in [0..n], n in [0..15]]; // _G. C. Greubel_, Apr 06 2021

%o (Sage)

%o def A061176(n,k): return sum((-1)^(k+j)*binomial(n,2*j)*binomial(n-2*j,k-j) for j in (0..k))

%o flatten([[A061176(n,k) for k in (0..n)] for n in (0..15)]) # _G. C. Greubel_, Apr 06 2021

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

%K sign,easy,tabl

%O 0,12

%A _Wolfdieter Lang_, Apr 20 2001