A091147 Expansion of (1-x-sqrt(1-2x-15x^2))/(8x^2).

1, 1, 5, 13, 57, 201, 861, 3445, 14897, 63313, 278389, 1223069, 5465065, 24513945, 111037005, 505298565, 2314343265, 10645982625, 49202944485, 228253816365, 1062783893145, 4964167491945, 23256852644925, 109249893866133, 514494575459217, 2428488338526961

Offset: 0

Author

Paul Barry, Dec 22 2003

Keywords

• easy
• nonn

Comments

a(n) = A014433(n+1)/4.

Number of lattice paths in the first quadrant from (0,0) to (n,0) using only steps H=(1,0), U=(1,1) and D=(1,-1), where the U steps come in 4 colors (i.e. Motzkin paths with the up steps in 4 colors). Series reversion of x/(1+x+4x^2). - Paul Barry, May 16 2005

Links

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

Crossrefs

Cf. A217275, A084605.

Programs

Maple

a := n -> simplify(2^n*GegenbauerC(n, -n-1, -1/4)/(n+1)):
seq(a(n), n=0..25); # Peter Luschny, May 08 2016

Mathematica

a[0] = 1; a[1] = 1; a[n_] := ((2*n + 1)*a[n - 1] - 15*(1 - n)*a[n - 2])/(n + 2); Table[a[n], {n, 0, 50}] (* T. D. Noe, Oct 02 2012 *)
CoefficientList[Series[(1 - x - Sqrt[1 - 2 x - 15 x^2]) / (8 x^2), {x, 0, 30}], x] (* Vincenzo Librandi, May 10 2013 *)
a[n_] := Hypergeometric2F1[1/2 - n/2, -n/2, 2, 16];
Table[a[n], {n, 0, 25}] (* Peter Luschny, Mar 18 2018 *)

Maxima

a(n):=2^n*coeff(expand((1+x/2+x^2)^(n+1)),x^n)/(n+1);
makelist(a(n),n,0,30); /* Emanuele Munarini, Apr 27 2012 */

PARI

my(x='x+O('x^66)); Vec((1-x-sqrt(1-2*x-15*x^2))/(8*x^2)) \\ Joerg Arndt, May 11 2013

Formula

G.f.: 2/(1-x+sqrt(1-2x-15x^2)).

G.f.: exp( Sum_{n>=1} A084605(n) * x^n/n ). - Paul D. Hanna, Dec 08 2018

a(n) = sum{k=0..n, binomial(n, k)4^(k/2)C(k/2)(1+(-1)^k)/2}, C(n)=A000108(n).

a(n) = sum{k=0..n, C(n, 2k)C(k)4^k}. - Paul Barry, May 16 2005

a(n) = integral(x=-2..2, (2*x+1)^n*sqrt((2-x)*(2+x)))/(2*Pi). [Peter Luschny, Sep 11 2011]

a(n) = (2^n/(n+1))*[x^n] (1+x/2+x^2)^(n+1). [Emanuele Munarini, Apr 27 2012]

E.g.f.: a(n) = n! * [x^n] exp(x)*BesselI(1, 4*x)/(2*x). -Peter Luschny, Aug 25 2012

D-finite with recurrence: (n+2)*a(n) -(2*n+1)*a(n-1) +15*(1-n)*a(n-2)=0. - R. J. Mathar, Sep 26 2012, [corrected by Vaclav Kotesovec, Sep 29 2012]

a(n) ~ 5/8*sqrt(10)/(n^(3/2)*sqrt(Pi))*5^n. - Vaclav Kotesovec, Sep 29 2012

a(n) = hypergeom([-n/2, (1-n)/2], [2], 16). - Peter Luschny, May 28 2014

a(n) = 2^n*GegenbauerC(n,-n-1, -1/4)/(n+1). - Peter Luschny, May 08 2016

G.f.: 1/(1 - x - 4*x^2/(1 - x - 4*x^2/(1 - x - 4*x^2/(1 - x - 4*x^2/(1 - ....))))), a continued fraction. - Ilya Gutkovskiy, May 26 2017