A171846 Triangle read by rows: T(n,k) is the number of peakless Motzkin paths of length n for which the area below the path is k (n>=0, k>=0).

1, 1, 1, 1, 0, 1, 1, 0, 2, 1, 1, 0, 3, 2, 1, 0, 1, 1, 0, 4, 3, 3, 1, 2, 2, 1, 1, 0, 5, 4, 6, 4, 4, 4, 5, 2, 1, 0, 1, 1, 0, 6, 5, 10, 9, 9, 7, 11, 8, 5, 3, 3, 2, 2, 1, 1, 0, 7, 6, 15, 16, 18, 14, 20, 20, 16, 10, 11, 8, 8, 6, 5, 2, 1, 0, 1, 1, 0, 8, 7, 21, 25, 32, 28, 36, 39, 41, 29, 27, 24, 25, 20, 17

Offset

0,9

Comments

Row 2n (n>0) has n^2 entries; row 2n+1 has n^2 + n + 1 entries.

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

Sum_{k>=0} k*T(n,k) = A171847(n).

Links

Table of n, a(n) for n=0..92.

Formula

G.f. G=G(t,z) satisfies: G(t,z) = 1/(1 - z + tz^2 - tz^2*G(t,tz)) (yielding a continued-fraction expression for G(t,z)).

Example

T(4,2)=2 because we have HUHD and UHDH, where U=(1,1), H=(1,0), D=(1,-1).
Triangle starts:
  1;
  1;
  1;
  1, 0, 1;
  1, 0, 2, 1;
  1, 0, 3, 2, 1, 0, 1;
  1, 0, 4, 3, 3, 1, 2, 2, 1;

Maple

g[0] := 1/(1-z+t*z^2-t*z^2*g[1]): for n to 15 do g[n] := subs({z = t*z, g[n] = g[n+1]}, g[n-1]) end do: G := subs(g[16] = 0, g[0]): Gser := simplify(series(G, z = 0, 15)): for n from 0 to 10 do P[n] := sort(coeff(Gser, z, n)) end do: d := proc (n) if n = 0 then 0 elif `mod`(n, 2) = 0 then (1/4)*(n-2)*(n+2) else (1/4)*(n-1)*(n+1) end if end proc: for n from 0 to 10 do seq(coeff(P[n], t, k), k = 0 .. d(n)) end do; # yields sequence in triangular form

Crossrefs

Cf. A004148, A171847.

Sequence in context: A055168 A085144 A156578 * A097230 A144789 A285097

Adjacent sequences: A171843 A171844 A171845 * A171847 A171848 A171849

Keyword

nonn,tabf

Author

Emeric Deutsch, Feb 08 2010

Extensions

Keyword tabf added by Michel Marcus, Apr 09 2013

Status

approved