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A123218 Irregular triangle formed by coefficients of polynomials defined by P(n,k,x) = f(n,k)*(2*x)^k*(1 - x^2)^(n - k), where f(n, k) = (-1)^floor((k + 1)/2)* binomial(n - floor((k + 1)/2), floor(k/2)).

%I A123218 #29 Apr 19 2019 11:03:41
%S A123218 1,1,-2,-1,1,-2,-6,2,1,1,-2,-11,12,11,-2,-1,1,-2,-16,22,46,-22,-16,2,
%T A123218 1,1,-2,-21,32,106,-92,-106,32,21,-2,-1,1,-2,-26,42,191,-212,-396,212,
%U A123218 191,-42,-26,2,1,1,-2,-31,52,301,-382,-1011,792,1011,-382,-301,52,31,-2,-1
%N A123218 Irregular triangle formed by coefficients of polynomials defined by  P(n,k,x) = f(n,k)*(2*x)^k*(1 - x^2)^(n - k), where f(n, k) = (-1)^floor((k + 1)/2)* binomial(n - floor((k + 1)/2), floor(k/2)).
%H A123218 G. C. Greubel, <a href="/A123218/b123218.txt">Rows n = 1..101 of triangle, flattened</a>
%H A123218 P. Steinbach, <a href="http://www.jstor.org/stable/2691048">Golden fields: a case for the heptagon</a>, Math. Mag. 70 (1997), no. 1, 22-31.
%F A123218 Let f(n, k) = (-1)^floor((k + 1)/2)*binomial(n - floor((k + 1)/2), floor(k/2)) then the polynomials P(n, k, x) = f(n,k)*(2*x)^k*(1 - x^2)^(n - k) for an irregular triangle of coefficients.
%e A123218 Triangle begins with:
%e A123218 1;
%e A123218 1, -2,  -1;
%e A123218 1, -2,  -6,  2,  1;
%e A123218 1, -2, -11, 12, 11,  -2,  -1;
%e A123218 1, -2, -16, 22, 46, -22, -16, 2, 1;
%t A123218 f[n_, k_]:= (-1)^Floor[(k+1)/2]*Binomial[n -Floor[(k+1)/2], Floor[k/2]]; Table[CoefficientList[Sum[f[n, k]*(2*x)^k*(1-x^2)^(n-k), {k, 0, n}], x], {n, 0, 10}]//Flatten
%Y A123218 Cf. A038223, A122600, A122753.
%K A123218 tabf,sign
%O A123218 1,3
%A A123218 _Roger L. Bagula_ and _Gary W. Adamson_, Oct 04 2006