

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.


3



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, 59, 61, 33, 10, 1, 51, 134, 147, 76, 15, 0, 85, 267, 327, 208, 75, 15, 1, 188, 584, 771, 528, 186, 26, 0, 344, 1209, 1734, 1329, 585, 150, 21, 1, 730, 2608, 4008, 3344, 1595, 408
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OFFSET

0,10


COMMENTS

Sum of entries in row n is the secondary structure number A004148(n1) (n>=2).
Row n contains 1 + floor(n/2) entries.
T(n,0) = A167638(n).
Sum_{k>=0} k*T(n,k) = A167639(n).


LINKS

Table of n, a(n) for n=0..69.


FORMULA

G.f.: G=G(t,z) satisfies z(1+ztz^2)G^2(1+zz^2)(1+ztz^2)G + 1+zz^2=0.
The trivariate g.f. G=G(t,s,z), where t marks oddlevel peaks, s marks evenlevel peaks, and z marks semilength, satisfies aG^2  bG + c = 0, where a = z(1+zsz^2), b=(1+ztz^2)(1+zsz^2), c=1+ztz^2.


EXAMPLE

T(6,2)=3 because we have U(UD)DUUU(UD)DDD, UUU(UD)DDDU(UD)D, and UUU(UD)DU(UD)DDD (the evenlevel peaks are shown between parentheses).
Triangle starts:
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;


MAPLE

eq := z*(1+zt*z^2)*G^2(1+zz^2)*(1+zt*z^2)*G+1+zz^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


CROSSREFS

Cf. A004148, A167634, A167638, A167639.
Sequence in context: A276554 A297323 A257654 * A109754 A220074 A059259
Adjacent sequences: A167634 A167635 A167636 * A167638 A167639 A167640


KEYWORD

nonn,tabf


AUTHOR

Emeric Deutsch, Nov 08 2009


STATUS

approved



