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Numbers that appear as binomial coefficients exactly 4 times.
5

%I #19 Aug 06 2024 02:18:31

%S 10,15,21,28,35,36,45,55,56,66,78,84,91,105,126,136,153,165,171,190,

%T 220,231,253,276,286,300,325,330,351,364,378,406,435,455,462,465,495,

%U 496,528,560,561,595,630,666,680,703,715,741,780,792,816,820

%N Numbers that appear as binomial coefficients exactly 4 times.

%C Let f(k) be the sequence of numbers that appear as binomial coefficients exactly k times:

%C f(1) = {2}.

%C f(2) = A137905.

%C f(3) appears to be A000984 \ {1, 2}: central binomial coefficients greater than 2.

%C f(4) = this sequence.

%C f(5) appears to be empty.

%C f(6) = A098565.

%C f(7) appears to be empty.

%C f(8) begins with 3003.

%H T. D. Noe, <a href="/A098564/b098564.txt">Table of n, a(n) for n = 1..1000</a>

%t binmax = 10^5; dm = 100; Clear[f]; f[m_] := f[m] = (Join[Table[Binomial[n, k], {n, 1, m}, {k, 1, n-1}], Table[Table[{Binomial[n, 1], Binomial[n, 2]}, {2}], {n, m+1, binmax}]] // Flatten // Tally // Select[#, #[[1]] <= binmax && #[[2]] == 4&]&)[[All, 1]] // Sort; f[dm]; f[m = 2*dm]; While[f[m] != f[m-dm], Print[m]; m = m+dm]; f[m] (* _Jean-François Alcover_, Mar 10 2014 *)

%Y Cf. A000984, A098565, A137905.

%K nonn

%O 1,1

%A _Paul D. Hanna_, Oct 27 2004