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 A101975 #9 Sep 09 2023 22:28:02
%S A101975 1,1,1,2,2,1,5,4,4,1,14,9,11,7,1,42,23,27,28,11,1,132,65,66,87,62,16,
%T A101975 1,429,197,170,239,250,122,22,1,1430,626,471,627,829,630,219,29,1,
%U A101975 4862,2056,1398,1656,2448,2553,1419,366,37,1,16796,6918,4381,4554,6803,8813,6979,2917,578,46,1
%N A101975 Triangle read by rows: number of Dyck paths of semilength n with k peaks after the first return (0 <= k < n).
%D A101975 Emeric Deutsch, Dyck path enumeration, Discrete Math., 204, 1999, 167-202.
%F A101975 T(n,0) = c(n-1), T(n,1) = Sum_{i=0..n-2} c(i), T(n,k) = Sum_{j=0..n-2} c(j)*binomial(n-1-j, k-1)*binomial(n-1-j, k)/(n-1-j) for k >= 2, where c(i) = binomial(2i, i)/(i+1) (i=0, 1, ...) are the Catalan numbers (A000108).
%e A101975 T(4,2)=4 because we have UD|(UD)U(UD)D, UD|U(UD)D(UD), UD|U(UD)(UD)D and
%e A101975 UUDD|(UD)(UD), where U=(1,1), D=(1,-1) (the two peaks after the first return | are shown between parentheses).
%e A101975 Triangle begins:
%e A101975 1;
%e A101975 1, 1;
%e A101975 2, 2, 1;
%e A101975 5, 4, 4, 1;
%e A101975 14, 9, 11, 7, 1;
%e A101975 42, 23, 27, 28, 11, 1;
%e A101975 ...
%p A101975 c:=n->binomial(2*n,n)/(n+1):T:=proc(n,k) if k=0 then c(n-1) elif k=1 then sum(c(i),i=0..n-2) else sum(c(j)*binomial(n-1-j,k-1)*binomial(n-1-j,k)/(n-1-j),j=0..n-2) fi end: for n from 1 to 11 do seq(T(n,k),k=0..n-1) od; # yields the sequence in triangular form
%Y A101975 Cf. A000108, A014137, A101974.
%K A101975 nonn,tabl
%O A101975 1,4
%A A101975 _Emeric Deutsch_, Dec 22 2004