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

Author

Emeric Deutsch, Nov 08 2009

Keywords

• nonn
• tabf

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).

Examples

			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;
  ...
		

Crossrefs

Cf. A004148, A167634, A167638, A167639.

Programs

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

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.