A190164 - OEIS

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

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OFFSET

0,12

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.

LINKS

Alois P. Heinz, Rows n = 0..200, flattened

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.

EXAMPLE

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:

0, 1;

0, 0, 1;

1, 0, 0, 1;

1, 2, 0, 0, 1;

1, 3, 3, 0, 0, 1;

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

# Alternative:

b:= proc(x, y, t) option remember; expand(`if`(y<0 or y>x, 0,

`if`(x=0, 1, b(x-1, y, false)*`if`(y::even, z, 1)+

`if`(t, 0, b(x-1, y-1, false))+b(x-1, y+1, true))))

end:

T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(n, 0, false)):

seq(T(n), n=0..12); # Alois P. Heinz, Feb 01 2026

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 *)

Flatten@CoefficientList[CoefficientList[Nest[ 1-z+z^2+z # (t-z+(1+z (#-1)) (1-t z+z^2)) &, 1 + O[z], 14], z], t] (* Oliver Seipel, Feb 10 2026 *)