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A182401 Number of paths from (0,0) to (n,0), never going below the x-axis, using steps U=(1,1), H=(1,0) and D=(1,-1), where the H steps come in five colors. 3
1, 5, 26, 140, 777, 4425, 25755, 152675, 919139, 5606255, 34578292, 215322310, 1351978807, 8550394455, 54419811354, 348309105300, 2240486766555, 14476490777175, 93914850905862, 611489638708140, 3994697746533171, 26175407271617955, 171991872078871311 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Number of 3-colored Schroeder paths from (0,0) to (2n+2,0) with no level steps H=(2,0) at even level. H-steps at odd levels are  colored with one of the three colors. Example: a(2)=5 because we have UUDD, UHD (3 choices) and UDUD. - José Luis Ramírez Ramírez, Apr 27 2015

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

FORMULA

a(n) = [x^n] (1+5*x+x^2)^(n+1)/(n+1).

a(n) = sum(binomial(n,2*k)*binomial(2*k,k)/(k+1)*5^(n-2*k),k=0..n/2).

G.f. (1-5*x-sqrt(1-10*x+21*x^2))/(2*x^2).

Conjecture: (n+2)*a(n) +5*(-2*n-1)*a(n-1) +21*(n-1)*a(n-2)=0. - R. J. Mathar, Jul 24 2012

a(n) ~ 7^(n+3/2)/(2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 20 2012

a(n) = A125906(n,0). - Philippe Deléham, Mar 04 2013

MATHEMATICA

CoefficientList[Series[(1-5*x-Sqrt[1-10*x+21*x^2])/(2*x^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 20 2012 *)

a[n_] := 5^n*Hypergeometric2F1[(1-n)/2, -n/2, 2, 4/25]; Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Feb 22 2013, after 2nd formula *)

PROG

(Maxima) a(n):=coeff(expand((1+5*x+x^2)^(n+1)), x^n)/(n+1);

makelist(a(n), n, 0, 30);

(PARI) x='x+O('x^66); Vec((1-5*x-sqrt(1-10*x+21*x^2))/(2*x^2)) \\ Joerg Arndt, Jun 02 2013

CROSSREFS

Cf. A002212, A005572, A257290

Sequence in context: A005573 A081911 A081187 * A104498 A045379 A053487

Adjacent sequences:  A182398 A182399 A182400 * A182402 A182403 A182404

KEYWORD

nonn

AUTHOR

Emanuele Munarini, Apr 27 2012

STATUS

approved

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Last modified May 22 13:18 EDT 2017. Contains 286872 sequences.