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

%I A061177 #7 Apr 06 2021 23:09:33
%S A061177 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,
%T A061177 -14,-7,7,8,0,-56,120,-120,56,0,-8,9,12,-126,288,-365,288,-126,12,9,
%U A061177 10,30,-228,540,-770,770,-540,228,-30,-10,11,55,-363,858
%N A061177 Coefficients of polynomials ( (1 -x +sqrt(x))^(n+1) - (1 -x -sqrt(x))^(n+1) )/(2*sqrt(x)).
%C A061177 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.
%H A061177 G. C. Greubel, <a href="/A061177/b061177.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A061177 T(n, k) = coefficient of x^k of ( (1 -x +sqrt(x))^(n+1) - (1 -x -sqrt(x))^(n+1) )/(2*sqrt(x)).
%F A061177 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.
%F A061177 Sum_{k=0..n} T(n, k) = (1 + (-1)^n)/2 = A059841(n). - _G. C. Greubel_, Apr 06 2021
%e A061177 The first few polynomials are:
%e A061177 pFo(0, x) = 1.
%e A061177 pFo(1, x) = 2 -  2*x.
%e A061177 pFo(2, x) = 3 -  5*x +  3*x^2.
%e A061177 pFo(3, x) = 4 -  8*x +  8*x^2 -  4*x^3.
%e A061177 pFo(4, x) = 5 - 10*x + 11*x^2 - 10*x^3 +  5*x^4.
%e A061177 pFo(5, x) = 6 - 10*x +  6*x^2 -  6*x^3 + 10*x^4 - 6*x^5.
%e A061177 Number triangle begins as:
%e A061177    1;
%e A061177    2,  -2;
%e A061177    3,  -5,    3;
%e A061177    4,  -8,    8,  -4;
%e A061177    5, -10,   11, -10,    5;
%e A061177    6, -10,    6,  -6,   10,  -6;
%e A061177    7,  -7,  -14,  29,  -14,  -7,    7;
%e A061177    8,   0,  -56, 120, -120,  56,    0,  -8;
%e A061177    9,  12, -126, 288, -365, 288, -126,  12,   9;
%e A061177   10,  30, -228, 540, -770, 770, -540, 228, -30, -10;
%t A061177 T[n_, k_]:= Sum[(-1)^(k-j)*Binomial[n+1, 2*j+1]*Binomial[n-2*j, k-j], {j,0,k}];
%t A061177 Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* _G. C. Greubel_, Apr 06 2021 *)
%o A061177 (Magma)
%o A061177 A061177:= func< n,k | (&+[(-1)^(k+j)*Binomial(n+1,2*j+1)*Binomial(n-2*j,k-j): j in [0..k]]) >;
%o A061177 [A061177(n,k): k in [0..n], n in [0..15]]; // _G. C. Greubel_, Apr 06 2021
%o A061177 (Sage)
%o A061177 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))
%o A061177 flatten([[A061177(n,k) for k in (0..n)] for n in (0..15)]) # _G. C. Greubel_, Apr 06 2021
%Y A061177 Cf. A059841, A060921, A061176 (companion triangle).
%K A061177 sign,easy,tabl
%O A061177 0,2
%A A061177 _Wolfdieter Lang_, Apr 20 2001