A094675
Coefficients of polynomial in x multiplying cosh(x) in the modified spherical Bessel function of the first kind i_n(x).
Original entry on oeis.org
1, 0, -3, 0, 1, 0, 15, 0, -10, 0, -105, 0, 1, 0, 105, 0, 945, 0, -21, 0, -1260, 0, -10395, 0, 1, 0, 378, 0, 17325, 0, 135135, 0, -36, 0, -6930, 0, -270270, 0, -2027025, 0, 1, 0, 990, 0, 135135, 0, 4729725, 0, 34459425, 0, -55, 0, -25740, 0, -2837835, 0, -91891800, 0
Offset: 0
Sinh(x)/x, (x*cosh(x) - sinh(x))/x^2, (-3*x*cosh(x) + (3 + x^2)*sinh(x))/x^3, ...
Triangle begins:
1, 0;
-3, 0;
1, 0, 15, 0;
-10, 0, -105, 0;
1, 0, 105, 0, 945, 0;
...
A334823
Triangle, read by rows, of Lambert's denominator polynomials related to convergents of tan(x).
Original entry on oeis.org
1, 1, 0, 3, 0, -1, 15, 0, -6, 0, 105, 0, -45, 0, 1, 945, 0, -420, 0, 15, 0, 10395, 0, -4725, 0, 210, 0, -1, 135135, 0, -62370, 0, 3150, 0, -28, 0, 2027025, 0, -945945, 0, 51975, 0, -630, 0, 1, 34459425, 0, -16216200, 0, 945945, 0, -13860, 0, 45, 0, 654729075, 0, -310134825, 0, 18918900, 0, -315315, 0, 1485, 0, -1
Offset: 0
Polynomials:
f(0, x) = 1;
f(1, x) = x;
f(2, x) = 3*x^2 - 1;
f(3, x) = 15*x^3 - 6*x;
f(4, x) = 105*x^4 - 45*x^2 + 1;
f(5, x) = 945*x^5 - 420*x^3 + 15*x;
f(6, x) = 10395*x^6 - 4725*x^4 + 210*x^2 - 1;
f(7, x) = 135135*x^7 - 62370*x^5 + 3150*x^3 - 28*x;
f(8, x) = 2027025*x^8 - 945945*x^6 + 51975*x^4 - 630*x^2 + 1.
Triangle of coefficients begins as:
1;
1, 0;
3, 0, -1;
15, 0, -6, 0;
105, 0, -45, 0, 1;
945, 0, -420, 0, 15, 0;
10395, 0, -4725, 0, 210, 0, -1;
135135, 0, -62370, 0, 3150, 0, -28, 0;
2027025, 0, -945945, 0, 51975, 0, -630, 0, 1.
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C := ComplexField();
T:= func< n, k| Round( i^k*Factorial(2*n-k)*(1+(-1)^k)/(2^(n-k+1)*Factorial(k)*Factorial(n-k)) ) >;
[T(n,k): k in [0..n], n in [0..10]];
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T:= (n, k) -> I^k*(2*n-k)!*(1+(-1)^k)/(2^(n-k+1)*(k)!*(n-k)!);
seq(seq(T(n, k), k = 0 .. n), n = 0 .. 10);
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(* First program *)
y[n_, x_]:= Sqrt[2/(Pi*x)]*E^(1/x)*BesselK[-n -1/2, 1/x];
f[n_, k_]:= Coefficient[((-I)^n/2)*(y[n, I*x] + (-1)^n*y[n, -I*x]), x, k];
Table[f[n, k], {n,0,10}, {k,n,0,-1}]//Flatten
(* Second program *)
Table[ I^k*(2*n-k)!*(1+(-1)^k)/(2^(n-k+1)*(k)!*(n-k)!), {n,0,10}, {k,0,n}]//Flatten
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[[ i^k*factorial(2*n-k)*(1+(-1)^k)/(2^(n-k+1)*factorial(k)*factorial(n-k)) for k in (0..n)] for n in (0..10)]
Showing 1-2 of 2 results.
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