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Start writing $$\operatorname{N}(n, k) = \frac{1}{n} {n \choose k} {n \choose k-1}=\frac{n \, \big(\Gamma (n)\big)^2}{\Gamma (k)\, \Gamma(k+1)\, \Gamma (n+1-k)\, \Gamma(n+2-k)}$$ and the same for $\... • 268k 5 votes Accepted ### Closed form for the recurrence$S_n = 1 + S_{n-1} + \frac{2}{n} S_{n-2}$, where$S_1=1$and$S_2=2$? I went for a power series expansion and found $$\sum_{n=1}^\infty S_n x^n = \frac{1-e^{-2x}}{2(1-x)^3}~.$$ From here you can use the cauchy product to gain an explicit description of the sequence. $$... • 456 4 votes ### how to find closed form for \int_0^{\frac{\pi}{2}} \ln(1+\sqrt{\sin x}) dx? Let \mathcal{I} denote the value of the following trigonometric integral:$$\mathcal{I}:=\int_{0}^{\frac{\pi}{2}}\mathrm{d}\theta\,\ln{\left(1+\sqrt{\sin{\left(\theta\right)}}\right)}\approx0.874888.... • 30.9k 3 votes Accepted ### How can I prove$(\star)$? - I nice property of$_2F_1\left(\frac{5}{6},1;\frac{11}{6};-t^6 \right)$By definition $$_2F_1\left(\frac{5}{6},1;\frac{11}{6};-t^6 \right)=\sum_{n=0}^\infty (-1)^n\,\frac{5}{6 n+5}\,t^{6n}$$ The integral does not make much problems $$I= \int_{t}^{\infty} \frac{x^4}{1+x^6}... • 268k 2 votes Accepted ### Series of confluent hypergeometric functions with one extra Pochhamer symbol The answer to the question is to write the confluent hypergeometric series in summation form. Then reduce as many, or join, components as one can to an index of the form (x)_{p \, n + q \, k} over ... • 26.6k 1 vote ### how to find closed form for \int_0^{\frac{\pi}{2}} \ln(1+\sqrt{\sin x}) dx? This is not an answer since it leads to the same problem. Using what @Тyma Gaidash suggested in comments$$I=\int_0^{\frac{\pi}{2}} \log\bigg(1+\sqrt{\sin (x)}\bigg)\, dx=\int_0^1\frac{\cos^{-1}(x^2)}{... • 268k 1 vote ### On a derivative of Appell's$F_1$function with respect to a parameter Using integral representations: $$F_1(2,\alpha,\alpha;3;x,y) =\int_0^1 2 t (1-t x)^{-\alpha } (1-t y)^{-\alpha } \, dt$$ then: $$\left.\frac{\partial}{\partial\alpha}F_1(2,\alpha,\alpha;3;x,y)\right|_{... • 4,512 1 vote Accepted ### On a derivative of Appell's F_1 function with respect to a parameter You don't have all contributions from the cases (n=0, m>0) and (n>0, m=0). If you just write out the Pochhammer symbols you get$$ \left.\frac{\partial}{\partial \alpha} \left((\alpha)_n(\alpha)... • 362 1 vote ### Find coefficients of the inverse of a matrix holding sum of binomial coefficients The text below is between a long comment and a partial answer. I searched the On-Line Encyclopedia of Integer Sequences for the coefficients of the "generating" polynomials. I expect that ... • 93.4k 1 vote ### Asymptotics of hyperrgeometric 2F1 for large integer parameters With$z = \cosh \zeta$,$\lambda = n$,$\alpha = \frac{1}{2}$,$\beta = \frac{1}{2}$, and$\gamma = 2$, the first result in$\S9$of G. N. Watson's paper Asymptotic expansions of hypergeometric ... • 33.3k 1 vote ### Closed form for the recurrence$S_n = 1 + S_{n-1} + \frac{2}{n} S_{n-2}$, where$S_1=1$and$S_2=2\$?
Too long for a comment. After reading the more than elegant solution provided by @M.E.W., consider $$S_n(a)=-\frac{1}{4} \sum_{k=0}^{n-1} \frac{a^{n-k}}{(n-k)!} (k+2)(k+1)$$ the summation is given in ...