A190167 Triangle read by rows: T(n,k) is the number of peakless Motzkin paths of length n having a total of k (1,0)-steps at levels 1,3,5,... .

1, 1, 1, 1, 1, 1, 2, 1, 2, 3, 2, 1, 4, 6, 4, 2, 1, 7, 12, 10, 5, 2, 1, 12, 24, 23, 14, 6, 2, 1, 21, 48, 52, 36, 18, 7, 2, 1, 38, 96, 115, 90, 51, 22, 8, 2, 1, 70, 193, 254, 217, 138, 68, 26, 9, 2, 1, 130, 388, 559, 522, 358, 196, 87, 30, 10, 2, 1, 243, 782, 1220, 1240, 926, 542, 264, 108, 34, 11, 2, 1

Offset: 0

Author

Emeric Deutsch, May 06 2011

Keywords

• nonn
• tabf

Comments

Row n has n-1 entries (n>=3).

Sum of entries in row n is A004148(n) (the RNA secondary structure numbers).

T(n,0)=A190168(n).

Sum(kT(n,k),k>=0)=A190169(n).

The trivariate g.f. H(t,s,z), where t (s) marks (1,0)-steps at even (odd) levels and z marks length, satisfies

z^2(1-tz+z^2)H^2-(1-tz+z^2)(1-sz+z^2)H+1-sz+z^2=0.

Examples

			T(5,2)=2 because we have huh'h'd and uh'h'dh, where u=(1,1), h=(1,0), d=(1,-1) (the odd-level h-steps are marked).
Triangle starts:
1;
1;
1;
1,1;
1,2,1;
2,3,2,1;
4,6,4,2,1;
7,12,10,5,2,1;
		

Crossrefs

Cf. A004148, A190164, A190168, A190169

Programs

Maple

eq:=z^2*(1-z+z^2)*G^2-(1-z+z^2)*(1-s*z+z^2)*G+1-s*z+z^2 = 0: g:= RootOf(eq, G): Gser:= simplify(series(g, z = 0, 17)): for n from 0 to 13 do P[n] := sort(expand(coeff(Gser, z, n))) end do: 1; 1; for n from 0 to 13 do seq(coeff(P[n], s, k), k = 0 .. n-2) end do; # yields sequence in triangular form

Formula

G.f. = G = G(s,z) satisfies the equation z^2*(1-z+z^2)G^2-(1-z+z^2)(1-sz+z^2)G+1-sz+z^2=0.