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.

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).

Links

Table of n, a(n) for n=0..66.

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

Cf. A004148, A167635, A167636, A167637.

Sequence in context: A188023 A077888 A319668 * A174169 A248174 A125095

Adjacent sequences: A167631 A167632 A167633 * A167635 A167636 A167637

Keyword

nonn,tabf

Author

Emeric Deutsch, Nov 08 2009

Status

approved