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How to solve this recurrence equation? $$A_{n,k}=(n+1-k)A_{n-1,k-1}+(k+1)A_{n-1,k+1},n=1,2,3,...;k=0,1,2,...$$ and $$A_{1,0}=1,A_{1,1}=1,A_{2,0}=1,A_{2,1}=2$$ When I studied the following formula, I found the recursive formula, but I couldn't solve it: $$g_n(x)=\sin^nx(-1)^{n-1}\frac{d^{n-1}}{dx^{n-1}}\left(\frac{1+\cos x}{\sin x}\right)\qquad(1)$$ \begin{align} g_1(x)&=1+\cos x\\ g_2(x)&=1+2\cos x+\cos ^2x\\ ...\\ g_n(x)&=\sum_{k=0}^{n}A_{n,k}\cos ^kx \end{align} Any help would be appreciated.

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The coefficients are listed in OEIS A198895 in reverse order. No closed form is given there, but a useful recurrence if your cos(x) are substituted by the variable x of the polynomials there.

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