A103769 Trinomial transform of central binomial coefficients A001405.

1, 4, 21, 123, 757, 4788, 30817, 200784, 1320093, 8740284, 58193673, 389233287, 2613338091, 17602627006, 118892784555, 804951501469, 5461228061541, 37120212399708, 252720891884473, 1723088114793535, 11763751150648785

Offset: 0

Author

Paul Barry, Feb 15 2005

Keywords

• easy
• nonn

Links

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

Crossrefs

Cf. A082760.

Programs

Mathematica

CoefficientList[Series[((3*x+1-(21*x^2-10*x+1)^(1/2))/(2*x*(3*x-4)*(7*x-1)))^(1/2), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 24 2012 *)

Formula

a(n) = sum_{k=0..2n} T(n,k)*C(k,floor(k/2)), where T(n,k) is given by A027907.

a(n) = sum_{k=0..n} sum_{j=0..n} C(n,j)*C(j,k)*C(j+k,floor((j+k)/2)).

G.f.: ((3*x+1-(21*x^2-10*x+1)^(1/2))/(2*x*(3*x-4)*(7*x-1)))^(1/2). - Mark van Hoeij, Nov 16 2011

Conjecture: n*(2n+1)*a(n) +2(-61n^2+57n-20)*a(n-1) +3*(205n^2-523*n+346) * a(n-2) -72*(n-2)*(16n-33)*a(n-3) +567*(n-2)*(n-3)*a(n-4)=0. - R. J. Mathar, Dec 14 2011

a(n) ~ 7^(n+1/2)/sqrt(5*Pi*n). - Vaclav Kotesovec, Oct 24 2012