A099324 Expansion of (1 + sqrt(1 + 4x))/(2(1 + x)).

1, 0, -1, 3, -8, 22, -64, 196, -625, 2055, -6917, 23713, -82499, 290511, -1033411, 3707851, -13402696, 48760366, -178405156, 656043856, -2423307046, 8987427466, -33453694486, 124936258126, -467995871776, 1757900019100, -6619846420552, 24987199492704, -94520750408708

Offset: 0

Author

Paul Barry, Oct 12 2004

Keywords

• easy
• sign

Comments

Binomial transform is A099323. Second binomial transform is A072100.

Hankel transform is A049347. - Paul Barry, Aug 10 2009

Links

• Robert Israel, Table of n, a(n) for n = 0..1668
• Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.

Crossrefs

Cf. A014138.

Programs

Maple

f:= gfun:-rectoproc({(2+4*n)*a(n)+(4+5*n)*a(n+1)+(n+2)*a(n+2), a(0) = 1, a(1) = 0}, a(n), remember):
map(f, [$0..50]); # Robert Israel, Mar 27 2018

Mathematica

CoefficientList[Series[(1+Sqrt[1+4x])/(2(1+x)),{x,0,40}],x] (* Harvey P. Dale, Jan 30 2014 *)

Formula

a(n) = Sum_{k=0..2n} (2*0^(2n-k)-1)*C(k,floor(k/2)). - Paul Barry, Aug 10 2009

|a(n+2)| = A091491(n+2,2). - Philippe Deléham, Nov 25 2009

G.f.: T(0)/(2+2*x), where T(k) = k+2 - 2*x*(2*k+1) + 2*x*(k+2)*(2*k+3)/T(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 27 2013

D-finite with recurrence: (2+4*n)*a(n) + (4+5*n)*a(n+1) + (n+2)*a(n+2) = 0. - Robert Israel, Mar 27 2018