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%I #21 Sep 26 2022 20:06:54
%S 1,0,1,1,-1,1,-1,3,-2,1,2,-5,6,-3,1,-3,10,-13,10,-4,1,5,-18,29,-26,15,
%T -5,1,-8,33,-60,65,-45,21,-6,1,13,-59,122,-151,125,-71,28,-7,1,-21,
%U 105,-241,338,-321,217,-105,36,-8,1,34,-185,468,-730,784,-609,350,-148,45,-9,1
%N Coefficients of polynomials B(x,n) = ((1+a+b)*x - c)*B(x,n-1) - a*b*B(x,n-2) where B(x,0) = 1, B(x,1) = x, a=-b, b=1, c=1.
%H G. C. Greubel, <a href="/A136531/b136531.txt">Rows n = 0..50 of the triangle, flattened</a>
%F G.f.: (1+y) / (1 + (1-x)*y - y^2). - _Kevin Ryde_, Sep 21 2022
%F From _G. C. Greubel_, Sep 22 2022: (Start)
%F T(n, k) = coefficients of i^n*(ChebyshevU(n, (x-1)/(2*i)) - i*ChebyshevU(n-1, (x-1)/(2*i))).
%F T(n, k) = coefficients of (-1)^n*( Fibonacci(n+1, 1-x) - Fibonacci(n, 1-x) ).
%F T(n, k) = i^(k-n-1)*(i*GegenbauerC(n-k, k+1, 1/(2*i)) - GegenbauerC(n-k-1, k+1, 1/(2*i))).
%F T(n, 0) = Fibonacci(1-n) = (-1)^n*A212804(n) = A039834(n-1).
%F T(n, 1) = (-1)^(n-1)*A010049(n), n >= 1.
%F T(n, 2) = (-1)^n*A055243(n-2), n >= 2.
%F T(n, n) = 1.
%F T(n, n-1) = -(n-1).
%F T(n, n-2) = A000217(n-1), n >= 2.
%F T(n, n-3) = -A008728(n-3), n >= 3.
%F Sum_{k=0..n-2} T(n, k) = A000027(n-1), n >= 2.
%F Sum_{k=0..n} T(n, k) = 1.
%F Sum_{k=0..floor(n/2)} T(n-k, k) = A151575(n) = (-1)^n*A078008(n). (End)
%e Triangle begins
%e k=0 k=1 k=2 k=3 k=4 k=5 k=6
%e n=0: 1;
%e n=1: 0, 1;
%e n=2: 1, -1, 1;
%e n=3: -1, 3, -2, 1;
%e n=4: 2, -5, 6, -3, 1;
%e n=5: -3, 10, -13, 10, -4, 1;
%e n=6: 5, -18, 29, -26, 15, -5, 1;
%t (* First program *)
%t a = -b; c = 1; b = 1;
%t B[x_, n_]:= B[x, n]= If[n<2, x^n, ((1+a+b)*x -c)*B[x, n-1] -a*b*B[x, n-2]];
%t Table[CoefficientList[B[x,n], x], {n,0,10}]//Flatten
%t (* Second program *)
%t B[x_, n_]:= (-1)^n*(Fibonacci[n+1, 1-x] - Fibonacci[n, 1-x]);
%t Table[CoefficientList[B[x, n], x], {n,0,16}]//Flatten (* _G. C. Greubel_, Sep 22 2022 *)
%o (Magma)
%o C<i> := ComplexField(); // T = A136531
%o T:= func< n,k | k eq n select 1 else Round(i^(k-n-1)*(i*Evaluate(GegenbauerPolynomial(n-k, k+1), 1/(2*i)) - Evaluate(GegenbauerPolynomial(n-k-1, k+1), 1/(2*i)))) >;
%o [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Sep 26 2022
%o (SageMath)
%o def T(n,k): # T = A136531
%o if k==n: return 1
%o else: return i^(k-n-1)*(i*gegenbauer(n-k, k+1, 1/(2*i)) - gegenbauer(n-k-1, k+1, 1/(2*i)))
%o flatten([[T(n,k) for k in range(n+1)] for n in range(12)]) # _G. C. Greubel_, Sep 26 2022
%Y Cf. A000027, A000217, A008728, A010049, A039834.
%Y Cf. A055243, A078008, A136526, A151575, A212804.
%K tabl,sign
%O 0,8
%A _Roger L. Bagula_, Mar 23 2008
%E Offset corrected by _Kevin Ryde_, Sep 21 2022
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