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

Author

Emanuele Munarini, Apr 27 2012

Keywords

• nonn
• easy

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

Examples

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

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.

Crossrefs

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

Programs

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 *)

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

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)