I actually dreamed about Herb a few months ago. In my dream, I told him about some recent mathematical discovery of mine (I no longer remember what) that I thought he'd like. He replied that, being dead, he no longer found math interesting. Jim On 8/13/12, Warren Smith <warren.wds@gmail.com> wrote:
Herb Wilf recently died. Here are two open problems he posed.
1. Is there a series for pi=a0+a1+a2+... where a(n) is a rational function of n and a(n-1), which converges faster than geometrically? Equivalently, is there a hypergeometric entire function F(z) with F(1)=pi?
2. Understand the asymptotic behavior of the sequence http://oeis.org/A019447 counting the number of terms in the expansion of an NxN toeplitz-matrix determinant.
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I think I see a way to solve problem #2 to the extent of proving it grows ultimately exponentially but NOT superexponentially. Argue the number of terms is upper bounded by the number of ways to put N indistinguishable balls into 2N-1 boxes where the Kth box must contain at most min(K, 2N-K) balls. This in turn is upper bounded by the unrestricted #ways to put N indistinguishable balls into 2N-1 boxes, which is (3N-1 choose 2N-2) = (3N-1 choose N+1), which is upper bounded by an exponential function of N, specifically C^N for any C>6.75.
You can also prove a lower bound on the #terms by simply chopping the NxN matrix into 13x13 diagonal blocks (ignoring rest of matrix) thus showing C >= 1338076^(1/13) > 2.9598.
This (finite upper and lower bounds above 1 on C) was the main qualitative thing Wilf wanted to know.
I'm sure it is possible to get tighter upper and lower bounds, though.
-- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)
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