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Identity for Power Series and Binomial Coefficients
(2) can be derived from (1) by subtracting 1, dividing by $r$ and taking the limit as $r\to 0$. 
Jan 26 
answered  Identity for Power Series and Binomial Coefficients 
Jan 20 
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Probability question involving drawing balls from an urn
One way to do this is with the transfer matrix method. (See, e.g., chapter 4 of Richard Stanley's Enumerative Combinatorics, volume 1.) The basic idea is that as you draw the balls, you keep track of the colors of the last two balls drawn. You can represent the possible transitions as edges in a directed graph with weights that keep track of the number of R's, B's, and triples of each type, so that the numbers you want can be obtained by extracting coefficients from powers of a 4 by 4 matrix, and from this matrix you can get a rational generating function for the numbers. 
Jan 11 
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Binomial coefficient identity
This identity is also a special case of Vandermonde's theorem. 
Dec 26 
answered  A recurrence relation on Catalan numbers 
Dec 8 
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Counting path generating sentences in a specific formal language
For an arbitrary Turing machine there is little or nothing that can be said. 
Dec 7 
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Nontrivial question about fibonacci numbers?
More generally, for $p\ne 2$ or 5, $F_{n+p} \equiv F_{n+1} \pmod p$ if $(p5)=1$ and $F_{n+p}\equiv F_{n1} \pmod p$ if $(p5) = 1$. (Here $F_{1}=1$.) 
Dec 5 
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A natural sum over multisets (expectation over multinomial)
The generating function $f(x)=\sum_n n^n x^n/n!$ is equal to $1/(1+W(x))$, where $W$ is the Lambert $W$function. The denominator has a zero at $x=1/e$ which should enable you to get a good approximation to the coefficients of $f(x)^k$ by standard techniques (see, e.g., Flajolet and Sedgewick's Analytic Combinatorics). 
Nov 30 
revised 
Dyck paths on rectangles
improved TeX formatting 
Nov 29 
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Number of Dyck paths with k returns and b peaks
See Emeric Deutsch, Dyck Path Enumeration, Discrete Mathematics 204 (1999) 167202, section 6.5, sciencedirect.com/science/article/pii/S0012365X98003719 
Nov 28 
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Inclusionpreserving bijection between subsets of cardinality k and nk
See also Curtis Greene and Daniel J. Kleitman. Strong versions of Sperner’s theorem. J. Combinatorial Theory Ser. A, 20(1):80–88, 1976. For similar decompositions of other posets, search for "symmetric chain decomposition". 
Nov 11 
awarded  Yearling 
Nov 2 
revised 
Combinatorial Morse functions and random permutations
added 138 characters in body 
Nov 1 
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Where can I buy Napier's Bones/Rods?
I put it in back quotes which works, but puts it in a gray box which I couldn't get rid of. 
Nov 1 
revised 
Where can I buy Napier's Bones/Rods?
added 6 characters in body 
Oct 31 
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What are some examples of interesting uses of the theory of combinatorial species?
OK, Joyal described his proof using species, but he used (implicitly) a nonnatural bijection; more precisely, he used the fact that the number of linear orders of a finite set is equal to the number of permutations (i.e., sets of cycles) of the set, but the corresponding species are not isomorphic. The proof may have been inspired by species, but I don't think it's a good example of what species are good for. There are other ways to deal with exponential generating functions; the real power of species (at least in enumeration) is its application in enumeration under group action. 
Oct 30 
answered  What are some examples of interesting uses of the theory of combinatorial species? 
Oct 30 
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What are some examples of interesting uses of the theory of combinatorial species?
But Joyal's proof does not use species. His proof may have been inspired by species, but his bijection is not natural (i.e., not functorial) — not that there's anything wrong with that. 
Oct 24 
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Finite series with reciprocal factorials
This hypergeometric expression says no more and no less than that the coefficient of $x^{k+2}$ in the sum is $$\frac{1}{(nk)!\,(k+2)}.$$ 
Oct 24 
awarded  Enlightened 