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a(n) = binomial(n, floor((n-8)/2)).
4

%I #25 May 25 2024 13:43:16

%S 0,0,0,0,0,0,0,0,1,1,10,11,66,78,364,455,1820,2380,8568,11628,38760,

%T 54264,170544,245157,735471,1081575,3124550,4686825,13123110,20030010,

%U 54627300,84672315,225792840

%N a(n) = binomial(n, floor((n-8)/2)).

%H Robert Israel, <a href="/A037958/b037958.txt">Table of n, a(n) for n = 0..3324</a>

%F Conjecture: -(n+9)*(n-8)*a(n) +2*(n)*a(n-1) +4*n*(n-1)*a(n-2)=0. - _R. J. Mathar_, Jul 26 2015, verified by _Robert Israel_, Jun 28 2018

%F From _Robert Israel_, Jun 28 2018: (Start)

%F E.g.f.: I_8(2*x)+I_9(2*x), where I_k is the modified Bessel function of the first kind and order k.

%F G.f.: 256*x^8/((1+sqrt(1-4*x^2))^8*sqrt(1-4*x^2)) + 512*x^9/((1+sqrt(1-4*x^2))^9*sqrt(1-4*x^2)). (End)

%p seq(binomial(n,floor((n-8)/2)), n=0..50); # _Robert Israel_, Jun 28 2018

%t Table[Binomial[n,Floor[(n-8)/2]],{n,0,40}] (* _Harvey P. Dale_, Jun 11 2013 *)

%o (GAP) List([0..40],n->Binomial(n,Int((n-8)/2))); # _Muniru A Asiru_, Jun 29 2018

%K nonn

%O 0,11

%A _N. J. A. Sloane_.