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A035929 Number of Dyck n-paths starting U^mD^m (an m-pyramid), followed by a pyramid-free Dyck path. 4
0, 1, 1, 1, 2, 6, 19, 61, 200, 670, 2286, 7918, 27770, 98424, 351983, 1268541, 4602752, 16799894, 61642078, 227239086, 841230292, 3126039364, 11656497518, 43601626146, 163561902392, 615183356156, 2319423532024, 8764535189296, 33187922345210, 125912855167740 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Hankel transform is -A128834. - Paul Barry, Jul 04 2009

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..500

J.-L. Baril, S. Kirgizov, The pure descent statistic on permutations, Preprint, 2016.

Paul Barry, Chebyshev moments and Riordan involutions, arXiv:1912.11845 [math.CO], 2019.

W. Kuszmaul, Fast Algorithms for Finding Pattern Avoiders and Counting Pattern Occurrences in Permutations, arXiv:1509.08216 [cs.DM], 2015.

Murray Tannock, Equivalence classes of mesh patterns with a dominating pattern, MSc Thesis, Reykjavik Univ., May 2016.

FORMULA

G.f.: A(x) satisfies A^2*(x^2-2*x+2) - A*(x+1) + x = 0.

The generating function can be written as x/(1-x) times that of A082989.

G.f.: (2*x)/(1+x+(1-x)*sqrt(1-4*x)) = 1/(1-x(1-x)/(1-x/(1-x/(1-x/(1-x/(1-x/(1-... (continued fraction). - Paul Barry, Jul 04 2009

From Gary W. Adamson, Jul 14 2011: (Start)

a(n), n>0; is the upper left term in M^(n-1), where M is the infinite square production matrix:

  1, 1, 0, 0, 0, 0, ...

  0, 1, 1, 0, 0, 0, ...

  1, 1, 1, 1, 0, 0, ...

  1, 1, 1, 1, 1, 0, ...

  1, 1, 1, 1, 1, 1, ...

  ... (End)

D-finite with recurrence: 2*n*a(n) +4*(-3*n+4)*a(n-1) +(19*n-44)*a(n-2) + (-13*n + 36)*a(n-3) +2*(2*n-7)*a(n-4)=0. - R. J. Mathar, Nov 24 2012

a(n) ~ 3 * 4^n / (25 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 12 2014

From Alexander Burstein, Aug 05 2017: (Start)

G.f: A = x/(1-(1-x)*x*C) = x*C/(1+x^2*C^2) = x*C^3/(1+2*x*C^3), where C is the g.f. of A000108.

A/x composed with x*C = g.f. of A165543, where A and C are as above. (End)

EXAMPLE

The a(5) = 6 cases are UUUUUDDDDD, UDUUUDUDDD, UDUUUDDUDD, UDUUDUUDDDD, UDUUDUDUDUDD and UUDDUUDUDD.

MAPLE

A:= proc(n) option remember; if n=0 then 0 else convert (series ((A(n-1)^2 *(x^2-2*x+2) +x)/ (x+1), x, n+1), polynom) fi end: a:= n-> coeff (A(n), x, n): seq (a(n), n=0..25); # Alois P. Heinz, Aug 23 2008

MATHEMATICA

CoefficientList[Series[2*x/(1+x+(1-x)*Sqrt[1-4*x]), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 12 2014 *)

PROG

(PARI) x='x+O('x^30); concat([0], Vec(2*x/(1+x+(1-x)*sqrt(1-4*x)))) \\ G. C. Greubel, Jan 15 2018

(MAGMA) /* Expansion */ Q:=Rationals(); R<x>:=PowerSeriesRing(Q, 30); R!(2*x/(1+x+(1-x)*Sqrt(1-4*x))); // G. C. Greubel, Jan 15 2018

CROSSREFS

Cf. A082989.

Sequence in context: A052975 A275943 A228180 * A071646 A114627 A289591

Adjacent sequences:  A035926 A035927 A035928 * A035930 A035931 A035932

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

Edited by Louis Shapiro, Feb 16 2005

Wrong g.f. removed by Vaclav Kotesovec, Feb 12 2014

STATUS

approved

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Last modified August 6 15:50 EDT 2020. Contains 336255 sequences. (Running on oeis4.)