This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A075760 #7 Mar 05 2025 21:18:30
%S A075760 36,1225,19600,41616,1413721,48024900,1631432881,55420693056,
%T A075760 1882672131025,63955431761796,2172602007770041,73804512832419600
%N A075760 Nontrivial binomial coefficients which are perfect powers (A001597).
%C A075760 Triangular-square numbers (A001110) are a subset, except for 0 and 1.
%C A075760 "For C(n,k) k>=4 and any l>=2 no solutions exist and this is what Erdos proved by an ingenious argument. ... C(50, 3) = 140^2 is the only solution for k = 3, l=2." page 13 of Aigner and Ziegler.
%D A075760 Martin Aigner and Gunter M. Ziegler, Proofs from THE BOOK, Second Edition, Springer-Verlag, Berlin, 2000, Chapter 3, "Binomial coefficients are (almost) never powers," pages 13-16.
%t A075760 f[n_] := Apply[ GCD, Last[ Transpose[ FactorInteger[n]]]]; a = {}; Do[ If[ f[n(n - 1)/2] > 1, a = Append[a, Binomial[n, 2]]]; If[ f[n(n - 1)*(n - 2)/6] > 1, a = Append[a, Binomial[n, 3]]], {n, 5, 1500000}]
%Y A075760 Cf. A001110.
%K A075760 nonn,more
%O A075760 1,1
%A A075760 _Robert G. Wilson v_, Oct 08 2002
%E A075760 a(10)-a(12) from _Sean A. Irvine_, Mar 05 2025