Conference rusure::math

290.0. "Repeated Binomial Coefficients" by TOOLS::STAN () Wed May 22 1985 22:58

Let C(m,n) denote the binomial coefficient m!/(n!(m-n)!).

An interesting question is: What integers are binomial coefficients
in multiple ways?

David Singmaster has investigated this problem and has found that
C(n+1,k+1)=C(n,k+2) is an identity if n=F[2i+2]F[2i+3]-1 and
k=F[2i]F[2i+3]-1 where F[i] are the Fibonacci numbers beginning with F[0]=0.

The only other non-trivial repeated binomial coefficients up to 2^48 are
the following:

	  120	= C(16,2)	= C(10,3)
	  210	= C(21,2)	= C(10,4)
	 1540	= C(56,2)	= C(22,3)
	 7140	= C(120,2)	= C(36,3)
	11628	= C(153,2)	= C(19,5)
	24310	= C(221,2)	= C(17,8)

Note also that 3003 is the only known integer with 8 or more representations.

   3003 = C(3003,3002) = C(3003,1) = C(78,76) = C(78,2)
        = C(15,10)     = C(15,5)   = C(14,8)  = C(14,6)

Anyone care to extend the search further?

			References
			----------
H.L. Abbott, P. Erdos, and D. Hanson, "On the number of times an integer
	occurs as a binomial coefficient", American Mathematical Monthly,
	81(1974)256-261.
Leroy F. Meyers, Problem 857, Crux Mathematicorum, 11(1985)84-85.
David Singmaster, "How often does an integer occur as a binomial coefficient?".
	American Mathematical Monthly, 78(1971)385-386.
David Singmaster, "Repeated binomial coefficients and Fibonacci numbers",
	Fibonacci Quarterly, 13(1975)295-298.