A103821 A Whitney transform of the central binomial coefficients A000984.

1, 3, 11, 43, 179, 771, 3395, 15171, 68515, 311907, 1428835, 6578531, 30414435, 141105251, 656588899, 3063038051, 14321092195, 67088405091, 314825048675, 1479654425187, 6963888239203, 32815960756835, 154813864252003

Offset: 0

Author

Paul Barry, Feb 16 2005

Keywords

• easy
• nonn

Comments

Partial sums of A006139. The Whitney transform maps the sequence with g.f. g(x) to that with g.f. (1/(1-x))g(x(1+x)).

Links

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

Programs

Mathematica

CoefficientList[Series[1/((1-x)*Sqrt[1-4*x-4*x^2]), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 17 2012 *)

Formula

G.f. : 1/((1-x)sqrt(1-4x-4x^2));

a(n)=sum{k=0..n, sum{i=0..n, C(k, i-k)}*C(2k, k)}.

Conjecture: n*a(n) +(2-5n)*a(n-1) +2*a(n-2)+4*(n-1)*a(n-3)=0. - R. J. Mathar, Dec 14 2011

a(n) ~ sqrt(34+23*sqrt(2))*(2+2*sqrt(2))^n/(7*sqrt(Pi*n)). - Vaclav Kotesovec, Oct 17 2012