[seqfan] Re: A conjecture on divisibility involving binomial coefficients

[pevnev at juno.com]

FYI

---------- Forwarded Message ----------
From: David Broadhurst <D.Broadhurst at open.ac.uk>
To: NMBRTHRY at LISTSERV.NODAK.EDU
Subject: Re: A conjecture on divisibility involving binomial coefficients
Date: Wed, 20 Apr 2011 09:22:24 -0500

Zhi-Wei Sun defined
s(n) = sum(k=0,n,f(k)*f(n-k))/((2*n-1)*binomial(3*n,n)),
with
f(k) = binomial(6*k,3*k)*binomial(3*k,k),
and guessed that s(n)^(1/n) tends to 64 as n tends to infinity.

Let
L = log(n) + 4*log(2) + 3*log(3) + gamma_Euler.
Then I conjecture that
2*Pi*n*sqrt(3*Pi*n)*s(n)/64^n =
L + 23*L/(72*n) + 1/(2*n) + 25*L/(128*n^2) + 31/(324*n^2) + O(L/n^3)
with further terms in the asymptotic expansion obtainable
from the recurrence relation given by Sun.

David Broadhurst




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