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URL: https://oeis.org/A091147

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A091147
Expansion of (1-x-sqrt(1-2*x-15*x^2))/(8*x^2).
8
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
0,3
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+4*x^2). - Paul Barry, May 16 2005
LINKS
FORMULA
G.f.: 2/(1-x+sqrt(1-2*x-15*x^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} binomial(n, 2*k)*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) dx. - 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
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 *)
PROG
(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
CROSSREFS
Sequence in context: A149551 A149552 A084136 * A351072 A149553 A149554
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Dec 22 2003
STATUS
approved