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.

1, 5, 26, 140, 777, 4425, 25755, 152675, 919139, 5606255, 34578292, 215322310, 1351978807, 8550394455, 54419811354, 348309105300, 2240486766555, 14476490777175, 93914850905862, 611489638708140, 3994697746533171, 26175407271617955, 171991872078871311

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

Paveł Szabłowski, Beta distributions whose moment sequences are related to integer sequences listed in the OEIS, Contrib. Disc. Math. (2024) Vol. 19, No. 4, 85-109. See p. 98.

Formula

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

a(n) = Sum_{k=0..floor(n/2)} (binomial(n,2*k)*binomial(2*k,k)/(k+1))*5^(n-2*k).

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

G.f.: 1/(1 - 5*x - x^2/(1 - 5*x - x^2/(1 - 5*x - x^2/(1 - 5*x - x^2/(1 - ...))))), a continued fraction. - Ilya Gutkovskiy, Sep 21 2017

From Seiichi Manyama, Jan 15 2024: (Start)

G.f.: (1/x) * Series_Reversion( x / (1+5*x+x^2) ).

a(n) = (1/(n+1)) * Sum_{k=0..n} 3^(n-k) * binomial(n+1,n-k) * binomial(2*k+2,k). (End)

From Peter Bala, Feb 03 2024: (Start)

G.f: 1/(1 - 3*x)*c(x/(1 - 3*x))^2 = 1/(1 - 7*x)*c(-x/(1 - 7*x))^2, where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108.

a(n) = Sum_{k = 0..n} 3^(n-k)*binomial(n, k)*Catalan(k+1).

a(n) = 3^n * hypergeom([3/2, -n], [3], -4/3).

a(n) = 7^n * Sum_{k = 0..n} (-7)^(-k)*binomial(n, k)*Catalan(k+1).

a(n) = 7^n * hypergeom([3/2, -n], [3], 4/7). (End)

Example

seq(3^n * simplify(hypergeom([3/2, -n], [3], -4/3)), n = 0..20); # Peter Bala, Feb 04 2024

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. A001006, A002212, A005572, A025230, A126120, A257290.

Sequence in context: A005573 A081911 A081187 * A363308 A104498 A045379

Adjacent sequences: A182398 A182399 A182400 * A182402 A182403 A182404

Keyword

nonn,easy

Author

Emanuele Munarini, Apr 27 2012

Status

approved