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 A167637 #9 Sep 12 2024 20:05:59
%S A167637 1,0,0,1,1,0,0,1,1,2,2,0,1,3,3,1,5,8,4,0,5,13,12,6,1,15,32,27,8,0,21,
%T A167637 59,61,33,10,1,51,134,147,76,15,0,85,267,327,208,75,15,1,188,584,771,
%U A167637 528,186,26,0,344,1209,1734,1329,585,150,21,1,730,2608,4008,3344,1595,408,42,0
%N A167637 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n, having no ascents and no descents of length 1, and having k peaks at even level.
%C A167637 Sum of entries in row n is the secondary structure number A004148(n-1) (n>=2).
%C A167637 Row n contains 1 + floor(n/2) entries.
%C A167637 T(n,0) = A167638(n).
%C A167637 Sum_{k>=0} k*T(n,k) = A167639(n).
%F A167637 G.f.: G=G(t,z) satisfies z(1+z-tz^2)G^2-(1+z-z^2)(1+z-tz^2)G + 1+z-z^2=0.
%F A167637 The trivariate g.f. G=G(t,s,z), where t marks odd-level peaks, s marks even-level peaks, and z marks semilength, satisfies aG^2 - bG + c = 0, where a = z(1+z-sz^2), b=(1+z-tz^2)(1+z-sz^2), c=1+z-tz^2.
%e A167637 T(6,2)=3 because we have U(UD)DUUU(UD)DDD, UUU(UD)DDDU(UD)D, and UUU(UD)DU(UD)DDD (the even-level peaks are shown between parentheses).
%e A167637 Triangle starts:
%e A167637 1;
%e A167637 0;
%e A167637 0, 1;
%e A167637 1, 0;
%e A167637 0, 1, 1;
%e A167637 2, 2, 0;
%e A167637 1, 3, 3, 1;
%e A167637 5, 8, 4, 0;
%e A167637 5, 13, 12, 6, 1;
%e A167637 ...
%p A167637 eq := z*(1+z-t*z^2)*G^2-(1+z-z^2)*(1+z-t*z^2)*G+1+z-z^2 = 0: G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 20)): for n from 0 to 16 do P[n] := sort(expand(coeff(Gser, z, n))) end do: for n from 0 to 16 do seq(coeff(P[n], t, j), j = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form
%Y A167637 Cf. A004148, A167634, A167638, A167639.
%K A167637 nonn,tabf
%O A167637 0,10
%A A167637 _Emeric Deutsch_, Nov 08 2009