A099327 Expansion of ((1-x)*sqrt(1+2x) + (1+x)*sqrt(1-2x))/(2*(1-2x)^(5/2)).

1, 5, 16, 45, 117, 291, 700, 1646, 3799, 8647, 19448, 43330, 95738, 210094, 458216, 994204, 2146955, 4617439, 9893376, 21128058, 44982486, 95510090, 202278376, 427425860, 901236582, 1896594966, 3983929680, 8354539156, 17492095604

Offset: 0

Author

Paul Barry, Oct 12 2004

Keywords

• easy
• nonn

Comments

The g.f. is transformed to 1/(1-x)^5 under the Chebyshev transformation A(x)->1/(1+x^2)A(x/(1+x^2)). Second binomial transform of the sequence with g.f. 1/c(-x)^3, where c(x) is the g.f. of the Catalan numbers A000108.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A099325, A099326.

Programs

Mathematica

CoefficientList[Series[((1-x)*Sqrt[1+2*x]+(1+x)*Sqrt[1-2*x])/(2*(1-2*x)^(5/2)), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 08 2014 *)

Formula

a(n) = Sum_{k=0..n} (k+1)*binomial(n, (n-k)/2)*binomial(k+4, 4)*(1+(-1)^(n-k))/(n+k+2).

D-finite with recurrence: n*(n-3)*a(n) + 2*(-n^2+6)*a(n-1) + 4*(n-1)*(n-5)*a(n-2) + 8*(n-1)*(n-2)*a(n-3) = 0. - R. J. Mathar, Nov 24 2012

a(n) ~ 2^(n+1/2) *n^(3/2) / (3*sqrt(Pi)) * (1 + 9/8*sqrt(2*Pi/n)). - Vaclav Kotesovec, Feb 08 2014