OFFSET
0,8
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
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.
EXAMPLE
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;
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 *)
PROG
(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)
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
CROSSREFS
KEYWORD
tabl,sign
AUTHOR
Roger L. Bagula, Mar 23 2008
EXTENSIONS
Offset corrected by Kevin Ryde, Sep 21 2022
STATUS
approved