A099326 Expansion of ((1-2x)*sqrt(1+2x) + sqrt(1-2x))/(2*(1-2x)^(5/2)).

1, 4, 11, 28, 67, 156, 354, 792, 1747, 3820, 8278, 17832, 38174, 81368, 172644, 365104, 769411, 1617228, 3389838, 7090440, 14797546, 30828424, 64106716, 133113168, 275967022, 571415416, 1181585564, 2440680592, 5035637212

Offset: 0

Author

Paul Barry, Oct 12 2004

Keywords

• easy
• nonn

Comments

a(n) = Sum_{k=0..n} (k+1)*binomial(n,(n-k)/2)*binomial(k+3,3)*(1+(-1)^(n-k))/(n+k+2). The g.f. is transformed to 1/(1-x)^4 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)^2, where c(x) is the g.f. of the Catalan numbers A000108.

0, 1, 4, 11, 28, ... is the image of the quarter-squares floor((n+1)^2/4) (A002620(n+1)) under the Riordan array ((1+2x)/sqrt(1-4x^2), x*c(x^2)). Hankel transform of A099326 has g.f. (1-x)/(1+x)^4. - Paul Barry, Oct 25 2007

Links

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

Crossrefs

Cf. A099325, A099327.

Programs

Mathematica

CoefficientList[Series[((1-2*x)*Sqrt[1+2*x]+Sqrt[1-2*x])/(2*(1-2*x)^(5/2)), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 12 2014 *)

Formula

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

a(n) = Sum_{k=0..n} C(n,k)*(floor((abs(n-2k) + 1)^2/4) + floor((abs(n-2k+1) + 1)^2/4)). - Paul Barry, Oct 25 2007

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

a(n) ~ n * 2^(n-1) * (1 + 2*sqrt(2/(Pi*n))). - Vaclav Kotesovec, Feb 12 2014