%I M5374
%S 1,120,210,1540,3003,7140,11628,24310,61218182743304701891431482520
%N Numbers that occur 5 or more times in Pascal's triangle.
%C The subject of a recent thread on sci.math. Apparently it has been known for many years that there are infinitely many numbers that occur at least 6 times in Pascal's triangle, namely the solutions to binomial(n,m1) = binomial{n1,m) given by n = F_{2k}F_{2k+1}; m = F_{2k1}F_{2k} where F_i is the ith Fibonacci number. The first of these outside the range of the existing database entry is {104 choose 39} = {103 choose 40} = 61218182743304701891431482520.  _Christopher E. Thompson_, Mar 09 2001
%C It may be that there are no terms that appear exactly 5 times in Pascal's triangle, in which case the title could be changed to "Numbers that occur 6 or more times in Pascal's triangle".  _N. J. A. Sloane_, Nov 24 2004
%C No other terms below 33*10^16 (_David W. Wilson_).
%C 61218182743304701891431482520 really is the next term. Weger shows this and I checked it.  _T. D. Noe_, Nov 15 2004
%C Blokhuis et al. show that there are no other solutions less than 10^60, nor within the first 10^6 rows of Pascal's triangle other than those given by the parametric solution mentioned above.  _Christopher E. Thompson_, Jan 19 2018
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 93, #47.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Aart Blokhuis, Andries Brouwer, Benne de Weger, <a href="http://math.colgate.edu/~integers/vol17.html">Binomial collisions and near collisions</a>, INTEGERS, Volume 17, Article A64, 2017 (also available as <a href="https://arxiv.org/abs/1707.06893">arXiv:1707.06893 [math.NT]</a>).
%H B. M. M. de Weger, <a href="http://hdl.handle.net/1765/1356">Equal binomial coefficients: some elementary considerations</a>, Econometric Institute Research Papers, No. EI 9536/B, 1995.
%H B. M. M. de Weger, <a href="http://dx.doi.org/10.1006/jnth.1997.2109">Equal binomial coefficients: some elementary considerations</a>, Journal of Number Theory, Volume 63, Issue 2, April 1997, Pages 373386.
%H R. K. Guy and V. Klee, <a href="http://www.jstor.org/stable/2316321">Monthly research problems</a>, 19691971, Amer. Math. Monthly, 78 (1971), 11131122.
%H David Singmaster, <a href="http://www.fq.math.ca/Scanned/134/singmaster.pdf">Repeated binomial coefficients and Fibonacci numbers</a>, Fibonacci Quarterly 13 (1975) 295298.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PascalsTriangle.html">Pascal's Triangle</a>
%Y Cf. A003016, A059233.
%Y Cf. A182237, A098565 (subsequence).
%K nonn
%O 1,2
%A _N. J. A. Sloane_
