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A136531
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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.
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2
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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, -5, 1, -8, 33, -60, 65, -45, 21, -6, 1, 13, -59, 122, -151, 125, -71, 28, -7, 1, -21, 105, -241, 338, -321, 217, -105, 36, -8, 1, 34, -185, 468, -730, 784, -609, 350, -148, 45, -9, 1
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OFFSET
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0,8
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LINKS
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FORMULA
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G.f.: (1+y) / (1 + (1-x)*y - y^2). - Kevin Ryde, Sep 21 2022
T(n, k) = coefficients of i^n*(ChebyshevU(n, (x-1)/(2*i)) - i*ChebyshevU(n-1, (x-1)/(2*i))).
T(n, k) = coefficients of (-1)^n*( Fibonacci(n+1, 1-x) - Fibonacci(n, 1-x) ).
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))).
T(n, 1) = (-1)^(n-1)*A010049(n), n >= 1.
T(n, 2) = (-1)^n*A055243(n-2), n >= 2.
T(n, n) = 1.
T(n, n-1) = -(n-1).
Sum_{k=0..n-2} T(n, k) = A000027(n-1), n >= 2.
Sum_{k=0..n} T(n, k) = 1.
Sum_{k=0..floor(n/2)} T(n-k, k) = A151575(n) = (-1)^n*A078008(n). (End)
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EXAMPLE
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Triangle begins
k=0 k=1 k=2 k=3 k=4 k=5 k=6
n=0: 1;
n=1: 0, 1;
n=2: 1, -1, 1;
n=3: -1, 3, -2, 1;
n=4: 2, -5, 6, -3, 1;
n=5: -3, 10, -13, 10, -4, 1;
n=6: 5, -18, 29, -26, 15, -5, 1;
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MATHEMATICA
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(* First program *)
a = -b; c = 1; b = 1;
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]];
Table[CoefficientList[B[x, n], x], {n, 0, 10}]//Flatten
(* Second program *)
B[x_, n_]:= (-1)^n*(Fibonacci[n+1, 1-x] - Fibonacci[n, 1-x]);
Table[CoefficientList[B[x, n], x], {n, 0, 16}]//Flatten (* G. C. Greubel, Sep 22 2022 *)
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PROG
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(Magma)
C<i> := ComplexField(); // T = A136531
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)))) >;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 26 2022
(SageMath)
if k==n: return 1
else: return i^(k-n-1)*(i*gegenbauer(n-k, k+1, 1/(2*i)) - gegenbauer(n-k-1, k+1, 1/(2*i)))
flatten([[T(n, k) for k in range(n+1)] for n in range(12)]) # G. C. Greubel, Sep 26 2022
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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