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