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

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 ceiling(n/2) entries (n >= 1).

T(n,0) = A167635(n).

Sum_{k>=0} k*T(n,k) = A167636(n).

Examples

			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;
		

Crossrefs

Cf. A004148, A167635, A167636, A167637.

Programs

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

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.