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 A138496 #7 Mar 01 2023 15:34:21 %S A138496 0,1,10,120,3003 %N A138496 Where record values occur in A003016. %C A138496 It appears that the record values are 0, 3, 4, 6, 8, 10, 12, ... %C A138496 From _M. F. Hasler_, Feb 16 2023: (Start) %C A138496 The numbers that appear 3 times in Pascal's triangle are the central binomial coefficients (A000984), except for the number 2 that is the only number to appear only once. For n = 1 there would be an infinite number of occurrences, but sequence A003016 counts only the occurrences of n in rows <= n so that n = 1 also gives 3. %C A138496 All C(n,k) with 1 < k < n/2 (in particular triangular numbers A000217) appear at least 4 times; see A098564 for those appearing exactly 4 times. %C A138496 Numbers that appear 5 or more times are quite rare, they are listed in A003015 with subsequence A098565 of those appearing exactly 6 times. %C A138496 They are mostly C(n,k) with 2 < k < n/2 which are also triangular numbers, but some are also of the form C(n+1,k) = C(n,k+1) with 3 < k < n/2, and a subsequence of these has n and k given in terms of Fibonacci numbers. (End) %o A138496 (PARI) m=-1; [n | n<-[0..9999], m < m = max(A003016(n), m)] \\ _M. F. Hasler_, Feb 16 2023 %o A138496 (Python) m=-1; [n for n in range(9999)if m < (m := max(A003016(n), m))] # _M. F. Hasler_, Feb 16 2023 %K A138496 nonn,more %O A138496 1,3 %A A138496 _Reinhard Zumkeller_, Mar 20 2008