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 A167634 #5 Nov 24 2019 23:48:13
%S A167634 1,0,1,0,1,2,0,0,3,1,5,2,1,1,10,5,1,14,12,9,2,7,38,27,9,1,43,60,57,22,
%T A167634 3,36,156,146,69,15,1,143,284,326,176,45,4,166,672,784,482,155,23,1,
%U A167634 509,1320,1780,1224,453,82,5,731,2981,4162,3160,1354,313,33,1,1915,6104
%N A167634 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 odd level.
%C A167634 Sum of entries in row n is the secondary structure number A004148(n-1) (n >= 2).
%C A167634 Row n contains ceiling(n/2) entries (n >= 1).
%C A167634 T(n,0) = A167635(n).
%C A167634 Sum_{k>=0} k*T(n,k) = A167636(n).
%F A167634 G.f.: G=G(t,z) satisfies z(1+z-z^2)G^2-(1+z-z^2)(1+z-tz^2)G + 1+z-tz^2=0.
%F A167634 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 A167634 T(5,1)=3 because we have UUDDUU(UD)DD, UU(UD)DDUUDD, and UUUU(UD)DDDD (the odd-level peaks are shown between parentheses).
%e A167634 Triangle starts:
%e A167634 1;
%e A167634 0;
%e A167634 1;
%e A167634 0, 1;
%e A167634 2, 0;
%e A167634 0, 3, 1;
%e A167634 5, 2, 1;
%e A167634 1, 10, 5, 1;
%e A167634 14, 12, 9, 2;
%p A167634 eq := z*(1+z-z^2)*G^2-(1+z-z^2)*(1+z-t*z^2)*G+1+z-t*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: 1; for n to 16 do seq(coeff(P[n], t, j), j = 0 .. ceil((1/2)*n)-1) end do; # yields sequence in triangular form
%Y A167634 Cf. A004148, A167635, A167636, A167637.
%K A167634 nonn,tabf
%O A167634 0,6
%A A167634 _Emeric Deutsch_, Nov 08 2009