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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.
3
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, 3, 36, 156, 146, 69, 15, 1, 143, 284, 326, 176, 45, 4, 166, 672, 784, 482, 155, 23, 1, 509, 1320, 1780, 1224, 453, 82, 5, 731, 2981, 4162, 3160, 1354, 313, 33, 1, 1915, 6104
OFFSET
0,6
COMMENTS
Sum of entries in row n is the secondary structure number A004148(n-1) (n >= 2).
Row n contains ceiling(n/2) entries (n >= 1).
T(n,0) = A167635(n).
Sum_{k>=0} k*T(n,k) = A167636(n).
FORMULA
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.
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.
EXAMPLE
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).
Triangle starts:
1;
0;
1;
0, 1;
2, 0;
0, 3, 1;
5, 2, 1;
1, 10, 5, 1;
14, 12, 9, 2;
MAPLE
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
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Nov 08 2009
STATUS
approved