A094674
Coefficients of polynomial in x multiplying sinh(x) in the modified spherical Bessel function of the first kind i_n(x).
Original entry on oeis.org
1, -1, 1, 0, 3, -6, 0, -15, 1, 0, 45, 0, 105, -15, 0, -420, 0, -945, 1, 0, 210, 0, 4725, 0, 10395, -28, 0, -3150, 0, -62370, 0, -135135, 1, 0, 630, 0, 51975, 0, 945945, 0, 2027025, -45, 0, -13860, 0, -945945, 0, -16216200, 0, -34459425, 1, 0, 1485, 0, 315315, 0, 18918900, 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;
-1;
1, 0, 3;
-6, 0, -15;
1, 0, 45, 0, 105;
-15, 0, -420, 0, -945;
...
A334824
Triangle, read by rows, of Lambert's numerator polynomials related to convergents of tan(x).
Original entry on oeis.org
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
Offset: 0
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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C := 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]];
-
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);
-
(* 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
-
[[ 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)]
Showing 1-2 of 2 results.
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