A136531 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.
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
Offset: 0
Examples
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;
Links
- G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Crossrefs
Programs
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Magma
C := 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
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Mathematica
(* 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 *) -
SageMath
def T(n,k): # T = A136531 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
Formula
G.f.: (1+y) / (1 + (1-x)*y - y^2). - Kevin Ryde, Sep 21 2022
From G. C. Greubel, Sep 22 2022: (Start)
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).
T(n, n-2) = A000217(n-1), n >= 2.
T(n, n-3) = -A008728(n-3), n >= 3.
Sum_{k=0..n-2} T(n, k) = A000027(n-1), n >= 2.
Sum_{k=0..n} T(n, k) = 1.
Extensions
Offset corrected by Kevin Ryde, Sep 21 2022