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 A102319 A mean binomial transform of the central binomial numbers. 1
 1, 2, 7, 26, 107, 462, 2065, 9438, 43811, 205622, 972917, 4631838, 22157525, 106406978, 512629551, 2476289106, 11989326771, 58163714118, 282662269717, 1375801775214, 6705710840657, 32724623955882, 159880046446611 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Second binomial transform of A082758 (with interpolated zeros). LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 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 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 *) PROG (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 CROSSREFS Sequence in context: A150567 A000151 A150568 * A367236 A006603 A080244 Adjacent sequences: A102316 A102317 A102318 * A102320 A102321 A102322 KEYWORD easy,nonn AUTHOR Paul Barry, Jan 04 2005 STATUS approved

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