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 A125177 #21 Jun 16 2022 09:17:38
%S A125177 1,1,1,2,2,1,5,4,3,1,14,9,7,4,1,42,23,16,11,5,1,132,65,39,27,16,6,1,
%T A125177 429,197,104,66,43,22,7,1,1430,626,301,170,109,65,29,8,1,4862,2056,
%U A125177 927,471,279,174,94,37,9,1,16796,6918,2983,1398,750,453,268,131,46,10,1,58786
%N A125177 Triangle read by rows: T(n,0)=C(2n,n)/(n+1) for n>=0; T(n,n+1)=0; T(n,k)=T(n-1,k)+T(n-1,k-1) for 1<=k<=n.
%C A125177 Column k (k>=1) starts with 0, followed by the partial sums of column k-1. Row sums yield A126221.
%C A125177 Indexing n and k from 1 instead of from 0, T(n,k) is the number of Dyck n-paths whose first peak is at height k and whose first component avoids DUU. A primitive Dyck path is one whose only return (to ground level) is at the end. The interior returns of a general Dyck path split the path into a list of primitive Dyck paths, called its components. For example, UUDDUD has components UUDD, UD and T(4,2) = 4 counts UUDUDUDD, UUDDUUDD, UUDDUDUD, UUDUDDUD (but not UUDUUDDD because its first component contains a DUU). - _David Callan_, Jan 17 2007
%C A125177 Riordan array (c(x),x/(1-x)), c(x) the g.f. of A000108. Equal to ((1-x)*c(x),x)*A007318. [_Paul Barry_, May 06 2009]
%H A125177 Seiichi Manyama, <a href="/A125177/b125177.txt">Table of n, a(n) for n = 0..10000</a>
%F A125177 G.f.: G(t,x)=(1-x)[1-sqrt(1-4x)]/[2x(1-x-tx)].
%F A125177 T(n,k) = Sum_{j=0..n} C(n-j,k)*if(j=0,0^j, A000108(j)-A000108(j-1)). [_Paul Barry_, May 06 2009]
%F A125177 T(n,k) = Sum_{i=0..n-k} binomial(n-i-1,n-k-i)*A000108(i). - _Vladimir Kruchinin_, Nov 03 2016
%e A125177 First few rows of the triangle are:
%e A125177 1;
%e A125177 1, 1;
%e A125177 2, 2, 1;
%e A125177 5, 4, 3, 1;
%e A125177 14, 9, 7, 4, 1;
%e A125177 42, 23, 16, 11, 5, 1;
%e A125177 ...
%e A125177 (5,3) = 16 = 7 + 9 = (4,3) + (4,2).
%e A125177 From _Paul Barry_, May 06 2009: (Start)
%e A125177 Production matrix is
%e A125177 1, 1,
%e A125177 1, 1, 1,
%e A125177 1, 0, 1, 1,
%e A125177 2, 0, 0, 1, 1,
%e A125177 4, 0, 0, 0, 1, 1,
%e A125177 9, 0, 0, 0, 0, 1, 1,
%e A125177 21, 0, 0, 0, 0, 0, 1, 1,
%e A125177 51, 0, 0, 0, 0, 0, 0, 1, 1,
%e A125177 127, 0, 0, 0, 0, 0, 0, 0, 1, 1 (End)
%p A125177 T:=proc(n,k) if k=0 then binomial(2*n,n)/(n+1) elif n=0 then 0 else T(n-1,k)+T(n-1,k-1) fi end: for n from 0 to 11 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form
%p A125177 G:=(1-x)*(1-sqrt(1-4*x))/2/x/(1-x-t*x): Gser:=simplify(series(G,x=0,15)): for n from 0 to 11 do P[n]:=sort(coeff(Gser,x,n)) od: for n from 0 to 11 do seq(coeff(P[n],t,j),j=0..n) od; # yields sequence in triangular form
%o A125177 (Maxima) T(n,k)=sum((binomial(2*i,i)*binomial(n-i-1,n-k-i))/(i+1),i,0,n-k); /* _Vladimir Kruchinin_, Nov 03 2016 */
%Y A125177 Cf. A000108, A125178, A126221.
%K A125177 nonn,tabl
%O A125177 0,4
%A A125177 _Gary W. Adamson_, Nov 22 2006
%E A125177 Edited by _Emeric Deutsch_, Dec 28 2006
%E A125177 Definition amended by _Georg Fischer_, Jun 16 2022