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A126216 Triangle read by rows: T(n,k) is the number of Schroeder paths of semilength n containing exactly k peaks but no peaks at level one (n>=1; 0<=k<=n-1). 8
1, 2, 1, 5, 5, 1, 14, 21, 9, 1, 42, 84, 56, 14, 1, 132, 330, 300, 120, 20, 1, 429, 1287, 1485, 825, 225, 27, 1, 1430, 5005, 7007, 5005, 1925, 385, 35, 1, 4862, 19448, 32032, 28028, 14014, 4004, 616, 44, 1, 16796, 75582, 143208, 148512, 91728, 34398, 7644, 936, 54 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

A Schroeder path of semilength n is a lattice path in the first quadrant, from the origin to the point (2n,0) and consisting of steps U=(1,1), D=(1,-1) and H=(2,0).

Also number of Schroeder paths of semilength n containing exactly k doublerises but no (2,0) steps at level 0 (n>=1; 0<=k<=n-1). Also number of doublerise-bicolored Dyck paths (doublerises come in two colors; called also marked Dyck paths) of semilength n and having k doublerises of a given color (n>=1; 0<=k<=n-1). Also number of 12312- and 121323-avoiding matchings on [2n] with exactly k crossings.

Essentially the triangle given by [1,1,1,1,1,1,1,1,...] DELTA [0,1,0,1,0,1,0,1,0,1,...] where DELTA is the operator defined in A084938 . - Philippe Deléham, Oct 20 2007

Mirror image of triangle A033282 . - Philippe Deléham, Oct 20 2007

For relation to Lagrange inversion, or series reversion and the geometry of associahedra, or Stasheff polytopes, (and other combinatorial objects) see A133437. [Tom Copeland, Sep 29 2008]

LINKS

Table of n, a(n) for n=1..54.

W. Y. C. Chen, T. Mansour and S. H. F. Yan, Matchings avoiding partial patterns, The Electronic Journal of Combinatorics 13, 2006, #R112, Theorem 3.3.

D. Callan, Polygon Dissections and Marked Dyck Paths

Jean-Christophe Novelli and Jean-Yves Thibon, Duplicial algebras and Lagrange inversion, arXiv preprint arXiv:1209.5959, 2012.

J.-C. Novelli, J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962, 2014. See Fig. 7.

FORMULA

T(n,k) = C(n,k)*C(2*n-k,n+1)/n (0<=k<=n-1).

G.f.: G(t,z)=(1-2*z-t*z-sqrt(1-4*z-2*t*z+t^2*z^2))/(2*(1+t)*z).

Equals N * P, where N = the Narayana triangle (A001263) and P = Pascal's triangle, as infinite lower triangular matrices. A126182 = P * N. - Gary W. Adamson, Nov 30 2007

G.f.: 1/(1-x-(x+xy)/(1-xy/(1-(x+xy)/(1-xy/(1-(x+xy)/(1-xy/(1-.... (continued fraction). [Paul Barry, Feb 06 2009]

Let h(t) = (1-t)^2/(1+(u-1)*(1-t)^2) = 1/(u+2*t+3*t^2+4*t^3+...), then a signed (n-1)-th row polynomial of A126216 is given by u^(2n-1)*(1/n!)*((h(t)*d/dt)^n) t, evaluated at t=0, with initial n=2. The power series expansion of h(t) is related to A181289. (cf. A086810) - Tom Copeland, Oct 09 2011

From Tom Copeland, Oct 10 2011: (Start)

With polynomials

P(0,t) = 0

P(1,t) = 1

P(2,t) = 1

P(3,t) = 2 + t

P(4,t) = 5 + 5 t + t^2

P(5,t) = 14 + 21 t + 9 t^2 + t^3

The o.g.f. A(x,t)= {1+x*t-sqrt[(1-x*t)^2-4x]}/[2(1+t)], and

B(x,t)=x-x^2/(1-t*x)=x-x^2-((t*x)^3+(t*x)^4+...) is the compositional inverse in x. (series corrected by Tom Copeland, Sep 16 2014)

Let h(x,t)= 1/(dB/dx)= (1-tx)^2/[1-(t+1)(2x-tx^2)] =1/[1-2x-3tx^2+4t^2x^3+...]. Then P(n,t)=(1/n!)(h(x,t)*d/dx)^n x, evaluated at x=0, A= exp(x*h(u,t)*d/du) u, evaluated at u=0, and  dA/dx = h(A(x,t),t). (End)

EXAMPLE

T(3,1)=5 because we have HUUDD, UDHH, UUUDDD, UHUDD and UUDHD.

Triangle starts:

1;

2,1;

5,5,1;

14,21,9,1;

42,84,56,14,1;

Triangle [1,1,1,1,1,1,1,...] DELTA [0,1,0,1,0,1,0,1,...] begins:

1;

1, 0;

2, 1, 0;

5, 5, 1, 0;

14, 21, 9, 1, 0;

42, 84, 56, 14, 1, 0 ;...

MAPLE

T:=(n, k)->binomial(n, k)*binomial(2*n-k, n+1)/n: for n from 1 to 11 do seq(T(n, k), k=0..n-1) od; # yields sequence in triangular form

CROSSREFS

Cf. A000108, A002054, A002055, A002056, A007160, A033280, A033281.

Cf. A126182.

Sequence in context: A123971 A060920 A107842 * A124733 A137597 A059340

Adjacent sequences:  A126213 A126214 A126215 * A126217 A126218 A126219

KEYWORD

nonn,tabl

AUTHOR

Emeric Deutsch, Dec 20 2006

STATUS

approved

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Last modified November 21 14:12 EST 2014. Contains 249779 sequences.