A102319 A mean binomial transform of the central binomial numbers.

1, 2, 7, 26, 107, 462, 2065, 9438, 43811, 205622, 972917, 4631838, 22157525, 106406978, 512629551, 2476289106, 11989326771, 58163714118, 282662269717, 1375801775214, 6705710840657, 32724623955882, 159880046446611

Offset: 0

Author

Paul Barry, Jan 04 2005

Keywords

• easy
• nonn

Comments

Second binomial transform of A082758 (with interpolated zeros).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Programs

Maple

A102319 := proc(n)
    add(binomial(n, k)*binomial(2*k, k)*(1+(-1)^(n-k))/2,k=0..n) ;
end proc: # R. J. Mathar, Feb 20 2015

Mathematica

CoefficientList[Series[(1/Sqrt[1-6*x+5*x^2]+1/Sqrt[1-2*x-3*x^2])/2, {x, 0, 20}], x] (* Vaclav Kotesovec, Sep 29 2013 *)

PARI

x='x+O('x^50); Vec((1/sqrt(1-6*x+5*x^2) + 1/sqrt(1-2*x-3*x^2))/2) \\ G. C. Greubel, Mar 16 2017

Formula

G.f.: (1/sqrt(1-6*x+5*x^2) + 1/sqrt(1-2*x-3*x^2))/2.

a(n) = Sum_{k=0..floor(n/2)} binomial(n,2*k)*binomial(2*(n-2*k), n-2*k).

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

E.g.f.: cosh(x)*exp(2*x)*I_0(2x). - Paul Barry, May 01 2005

a(n) ~ 5^(n+1/2)/(4*sqrt(Pi*n)). - Vaclav Kotesovec, Sep 29 2013

Conjecture: n*(n-1)*a(n) -4*(n-1)*(3*n-4)*a(n-1) +(53*n^2-221*n+232)*a(n-2) +8*(-13*n^2+85*n-134)*a(n-3) +(51*n^2-563*n+1308)*a(n-4) +4*(29*n-93)*(n-4)*a(n-5) -105*(n-4)*(n-5)*a(n-6)=0. - R. J. Mathar, Feb 20 2015

Conjecture:+n*(n-1)*(12*n^2-48*n+41)*a(n) -8*(n-1)*(12*n^3-54*n^2+65*n-17)*a(n-1) +2*(84*n^4-504*n^3+1025*n^2-775*n+131)*a(n-2) +8*(n-2)*(12*n^3-54*n^2+65*n-20)*a(n-3) -15*(n-2)*(n-3)*(12*n^2-24*n+5)*a(n-4)=0. - R. J. Mathar, Feb 20 2015