cp's OEIS Frontend

A102319 A mean binomial transform of the central binomial numbers.

%I A102319 #20 Aug 31 2024 19:21:14
%S A102319 1,2,7,26,107,462,2065,9438,43811,205622,972917,4631838,22157525,
%T A102319 106406978,512629551,2476289106,11989326771,58163714118,282662269717,
%U A102319 1375801775214,6705710840657,32724623955882,159880046446611
%N A102319 A mean binomial transform of the central binomial numbers.
%C A102319 Second binomial transform of A082758 (with interpolated zeros).
%H A102319 G. C. Greubel, <a href="/A102319/b102319.txt">Table of n, a(n) for n = 0..1000</a>
%F A102319 G.f.: (1/sqrt(1-6*x+5*x^2) + 1/sqrt(1-2*x-3*x^2))/2.
%F A102319 a(n) = Sum_{k=0..floor(n/2)} binomial(n,2*k)*binomial(2*(n-2*k), n-2*k).
%F A102319 a(n) = Sum_{k=0..n} binomial(n,k)*binomial(2*k,k)*(1+(-1)^(n-k))/2.
%F A102319 E.g.f.: cosh(x)*exp(2*x)*I_0(2x). - _Paul Barry_, May 01 2005
%F A102319 a(n) ~ 5^(n+1/2)/(4*sqrt(Pi*n)). - _Vaclav Kotesovec_, Sep 29 2013
%F A102319 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
%F A102319 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
%p A102319 A102319 := proc(n)
%p A102319     add(binomial(n, k)*binomial(2*k, k)*(1+(-1)^(n-k))/2,k=0..n) ;
%p A102319 end proc: # _R. J. Mathar_, Feb 20 2015
%t A102319 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 *)
%o A102319 (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
%K A102319 easy,nonn
%O A102319 0,2
%A A102319 _Paul Barry_, Jan 04 2005