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.

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, 42, 0

Offset

0,10

Comments

Sum of entries in row n is the secondary structure number A004148(n-1) (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..71.

Formula

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.

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(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).
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+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

Crossrefs

Cf. A004148, A167634, A167638, A167639.

Sequence in context: A276554 A297323 A257654 * A343489 A360742 A109754

Adjacent sequences: A167634 A167635 A167636 * A167638 A167639 A167640

Keyword

nonn,tabf

Author

Emeric Deutsch, Nov 08 2009

Status

approved