

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<=n1).


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, 1
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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<=n1). Also number of doublerisebicolored 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<=n1). Also number of 12312 and 121323avoiding 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..55.
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
T. Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms, 2015
JeanChristophe Novelli and JeanYves Thibon, Duplicial algebras and Lagrange inversion, arXiv preprint arXiv:1209.5959 [math.CO], 2012.
J.C. Novelli, J.Y. Thibon, Hopf Algebras of mpermutations,(m+1)ary trees, and mparking functions, arXiv preprint arXiv:1403.5962 [math.CO], 2014. See Fig. 7.


FORMULA

T(n,k) = C(n,k)*C(2*nk,n+1)/n (0<=k<=n1).
G.f.: G(t,z)=(12*zt*zsqrt(14*z2*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/(1x(x+xy)/(1xy/(1(x+xy)/(1xy/(1(x+xy)/(1xy/(1.... (continued fraction). [Paul Barry, Feb 06 2009]
Let h(t) = (1t)^2/(1+(u1)*(1t)^2) = 1/(u+2*t+3*t^2+4*t^3+...), then a signed (n1)th row polynomial of A126216 is given by u^(2n1)*(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*tsqrt[(1x*t)^24x]}/[2(1+t)], and
B(x,t)=xx^2/(1t*x)=xx^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)= (1tx)^2/[1(t+1)(2xtx^2)] =1/[12x3tx^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*nk, n+1)/n: for n from 1 to 11 do seq(T(n, k), k=0..n1) od; # yields sequence in triangular form


MATHEMATICA

Table[Binomial[n, k] Binomial[2 n  k, n + 1]/n, {n, 10}, {k, 0, n  1}] // Flatten (* Michael De Vlieger, Jan 09 2016 *)


PROG

(PARI) tabl(nn) = {mP = matrix(nn, nn, n, k, binomial(n1, k1)); mN = matrix(nn, nn, n, k, binomial(n1, k1) * binomial(n, k1) / k); mprod = mN*mP; for (n=1, nn, for (k=1, n, print1(mprod[n, k], ", "); ); print(); ); } \\ Michel Marcus, Apr 16 2015


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



