

A104553


Sum of trapezoid weights of all Schroeder paths of length 2n.


2



1, 7, 38, 198, 1039, 5533, 29852, 162716, 893997, 4942723, 27466082, 153264066, 858230875, 4820155001, 27141345912, 153168964216, 866086326425, 4905744855359, 27830459812830, 158102366711550, 899290473825511, 5120997554408597, 29191620055374228, 166560724629655188
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OFFSET

1,2


COMMENTS

A Schroeder path is a lattice path starting from (0,0), ending at a point on the xaxis, consisting only of steps U=(1,1), D=(1,1) and H=(2,0) and never going below the xaxis. Schroeder paths are counted by the large Schroeder numbers (A006318). A trapezoid in a Schroeder path is a factor of the form U^i H^j D^i (i>=1, j>=0), i being the height of the trapezoid. A trapezoid in a Schroeder path w is maximal if, as a factor in w, it is not immediately preceded by a U and immediately followed by a D. The trapezoid weight of a Schroeder path is the sum of the heights of its maximal trapezoids. For example, in the Schroeder path w=UH(UHD)D(UUDD) we have two trapezoids (shown between parentheses) of heights 1 and 2, respectively. The trapezoid weight of w is 1+2=3. This concept is an analogous to the concept of pyramid weight in a Dyck path (see the DeniseSimion paper). Partial sums of A047665 which, in turn, are the partial sums of A002002.


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..200
A. Denise and R. Simion, Two combinatorial statistics on Dyck paths, Discrete Math., 137, 1995, 155176.


FORMULA

G.f.: (1xsqrt(16*x+x^2))/(2*(1x)^2*sqrt(16*x+x^2)).
Recurrence: n*(2*n3)*a(n) = 2*(8*n^2  15*n + 5)*a(n1)  2*(14*n^2  28*n + 11)*a(n2) + 2*(8*n^2  17*n + 7)*a(n3)  (n2)*(2*n1)*a(n4).  Vaclav Kotesovec, Oct 24 2012
a(n) ~ sqrt(48+34*sqrt(2))*(3+2*sqrt(2))^n/(16*sqrt(Pi*n)).  Vaclav Kotesovec, Oct 24 2012


EXAMPLE

a(2) = 7 because the six Schroeder paths of length 4, namely HH,(UD)H,H(UD),(UHD), (UD)(UD) and (UUDD), have trapezoid weights 0,1,1,1,2 and 2, respectively; the maximal trapezoids are shown between parentheses.


MAPLE

G:=(1zsqrt(16*z+z^2))/2/(1z)^2/sqrt(16*z+z^2):Gser:=series(G, z=0, 28): seq(coeff(Gser, z^n), n=1..25);


MATHEMATICA

CoefficientList[Series[(1  x  Sqrt[1  6 x + x^2]) / x /(2 (1  x)^2 Sqrt[1  6 x + x^2]), {x, 0, 30}], x] (* Harvey P. Dale, May 26 2011 *)


PROG

(PARI) x='x+O('x^66); Vec((1xsqrt(16*x+x^2))/(2*(1x)^2*sqrt(16*x+x^2))) \\ Joerg Arndt, May 13 2013


CROSSREFS

Cf. A006318, A047665, A002002, A104552.
Sequence in context: A141845 A048437 A099461 * A027241 A292761 A226200
Adjacent sequences: A104550 A104551 A104552 * A104554 A104555 A104556


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Mar 14 2005


EXTENSIONS

Typo in Mma program fixed by Vincenzo Librandi, May 13 2013


STATUS

approved



