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A334824
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Triangle, read by rows, of Lambert's numerator polynomials related to convergents of tan(x).
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2
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1, 3, 0, 15, 0, -1, 105, 0, -10, 0, 945, 0, -105, 0, 1, 10395, 0, -1260, 0, 21, 0, 135135, 0, -17325, 0, 378, 0, -1, 2027025, 0, -270270, 0, 6930, 0, -36, 0, 34459425, 0, -4729725, 0, 135135, 0, -990, 0, 1, 654729075, 0, -91891800, 0, 2837835, 0, -25740, 0, 55, 0, 13749310575, 0, -1964187225, 0, 64324260, 0, -675675, 0, 2145, 0, -1
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OFFSET
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0,2
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COMMENTS
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Lambert's denominator polynomials related to convergents of tan(x), f(n, x), are given in A334823.
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LINKS
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FORMULA
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Equals the coefficients of the polynomials, g(n, x), defined by: (Start)
g(n, x) = Sum_{k=0..floor(n/2)} ((-1)^k*(2*n-2*k+1)!/((2*k+1)!*(n-2*k)!))*(x/2)^(n-2*k).
g(n, x) = ((2*n+1)!/n!)*(x/2)^n*Hypergeometric2F3(-n/2, (1-n)/2; 3/2, -n, -n-1/2; -1/x^2).
g(n, x) = ((-i)^n/2)*(y(n+1, i*x) + (-1)^n*y(n+1, -i*x)), where y(n, x) are the Bessel Polynomials.
g(n, x) = (2*n-1)*x*g(n-1, x) - g(n-2, x).
E.g.f. of g(n, x): sin((1 - sqrt(1-2*x*t))/2)/sqrt(1-2*x*t).
g(n, 1) = (-1)^n*g(n, -1) = A053984(n) = (-1)^n*A053983(-n-1) = (-1)^n*f(-n-1, 1).
g(n, 2) = (-1)^n*g(n, -2) = A053987(n+1). (End)
As a number triangle:
T(n, k) = i^k*(2*n-k+1)!*(1+(-1)^k)/(2^(n-k+1)*(k+1)!*(n-k)!), where i = sqrt(-1).
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EXAMPLE
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Polynomials:
g(0, x) = 1;
g(1, x) = 3*x;
g(2, x) = 15*x^2 - 1;
g(3, x) = 105*x^3 - 10*x;
g(4, x) = 945*x^4 - 105*x^2 + 1;
g(5, x) = 10395*x^5 - 1260*x^3 + 21*x;
g(6, x) = 135135*x^6 - 17325*x^4 + 378*x^2 - 1;
g(7, x) = 2027025*x^7 - 270270*x^5 + 6930*x^3 - 36*x.
Triangle of coefficients begins as:
1;
3, 0;
15, 0, -1;
105, 0, -10, 0;
945, 0, -105, 0, 1;
10395, 0, -1260, 0, 21, 0;
135135, 0, -17325, 0, 378, 0, -1;
2027025, 0, -270270, 0, 6930, 0, -36, 0.
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MAPLE
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T:= (n, k) -> I^k*(2*n-k+1)!*(1+(-1)^k)/(2^(n-k+1)*(k+1)!*(n-k)!);
seq(seq(T(n, k), k = 0..n), n = 0..10);
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MATHEMATICA
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(* First program *)
y[n_, x_]:= Sqrt[2/(Pi*x)]*E^(1/x)*BesselK[-n -1/2, 1/x];
g[n_, k_]:= Coefficient[((-I)^n/2)*(y[n+1, I*x] + (-1)^n*y[n+1, -I*x]), x, k];
Table[g[n, k], {n, 0, 10}, {k, n, 0, -1}]//Flatten
(* Second program *)
Table[I^k*(2*n-k+1)!*(1+(-1)^k)/(2^(n-k+1)*(k+1)!*(n-k)!), {n, 0, 10}, {k, 0, n}]//Flatten
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PROG
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(Magma)
C<i> := ComplexField();
T:= func< n, k| Round( i^k*Factorial(2*n-k+1)*(1+(-1)^k)/(2^(n-k+1)*Factorial(k+1)*Factorial(n-k)) ) >;
[T(n, k): k in [0..n], n in [0..10]];
(Sage) [[ i^k*factorial(2*n-k+1)*(1+(-1)^k)/(2^(n-k+1)*factorial(k+1)*factorial(n-k)) for k in (0..n)] for n in (0..10)]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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