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### 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 ...
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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 $$\left.\frac{\partial}{\partial \alpha} \left((\alpha)_n(\alpha)... • 437 2 votes Accepted ### Series of confluent hypergeometric functions with one extra Pochhamer symbol The answer to the question is to write the confluent hypergeometric series in summation form. Then reduce as many, or join, components as one can to an index of the form (x)_{p \, n + q \, k} over ... • 26.6k 1 vote ### Asymptotics of hyperrgeometric 2F1 for large integer parameters With z = \cosh \zeta, \lambda = n, \alpha = \frac{1}{2}, \beta = \frac{1}{2}, and \gamma = 2, the first result in \S9 of G. N. Watson's paper Asymptotic expansions of hypergeometric ... • 33.3k 0 votes Accepted ### Card Game Opening Hand Calculator with subgroups Using conditional probability HYPGEOMDIST(2,5,3,40)=0,035 HYPGEOMDIST(3,5,3,40)=0,001 Now let's calculate the conditional probability of drawing exactly 2 ... • 1,060 1 vote ### Closed form for the recurrence S_n = 1 + S_{n-1} + \frac{2}{n} S_{n-2}, where S_1=1 and S_2=2? Too long for a comment. After reading the more than elegant solution provided by @M.E.W., consider$$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 ...
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