A190164 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 0,2,4,... .

1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 2, 0, 0, 1, 1, 3, 3, 0, 0, 1, 2, 4, 6, 4, 0, 0, 1, 4, 8, 9, 10, 5, 0, 0, 1, 7, 18, 19, 16, 15, 6, 0, 0, 1, 12, 35, 48, 36, 25, 21, 7, 0, 0, 1, 22, 66, 102, 100, 60, 36, 28, 8, 0, 0, 1, 41, 132, 209, 229, 180, 92, 49, 36, 9, 0, 0, 1, 76, 266, 450, 504, 440, 294, 133, 64, 45, 10, 0, 0, 1

Offset: 0

Author

Emeric Deutsch, May 06 2011

Keywords

• nonn
• tabl

Comments

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

T(n,0)=A190165(n).

Sum_{k>=0} k*T(n,k) = A190166(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 the equation

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)=3 because we have h'h'uhd, h'uhdh', and uhdh'h', where u=(1,1), h=(1,0), d=(1,-1) (the even-level h-steps are marked).
Triangle starts:
  1;
  0, 1;
  0, 0, 1;
  1, 0, 0, 1;
  1, 2, 0, 0, 1;
  1, 3, 3, 0, 0, 1;
		

Crossrefs

Cf. A004148, A190165, A190166, A110236, A190167.

Programs

Maple

eq := z^2*(1-t*z+z^2)*G^2-(1-z+z^2)*(1-t*z+z^2)*G+1-z+z^2 = 0: g := RootOf(eq, G): Gser := simplify(series(g, z = 0, 15)): for n from 0 to 13 do P[n] := sort(expand(coeff(Gser, z, n))) end do: for n from 0 to 12 do seq(coeff(P[n], t, k), k = 0 .. n) end do; # yields sequence in triangular form

Mathematica

m = 13; G[_] = 0;
Do[G[z_] = -((z^2 G[z]^2 (-t z + z^2 + 1) + z^2 - z + 1)/((z^2 - z + 1)(t z - z^2 - 1))) + O[z]^m, {m}];
CoefficientList[#, t]& /@ CoefficientList[G[z], z] // Flatten (* Jean-François Alcover, Nov 15 2019 *)

Formula

G.f.: G = G(t,z) satisfies the equation z^2*(1-tz+z^2)*G^2 - (1-z+z^2)*(1-tz+z^2)*G + 1 - z + z^2 = 0.