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a(n) = (3*0^n + 4^n*binomial(2*n,n))/4.
4

%I #15 Sep 08 2022 08:45:15

%S 1,2,24,320,4480,64512,946176,14057472,210862080,3186360320,

%T 48432676864,739699064832,11342052327424,174493112729600,

%U 2692179453542400,41639042214789120,645405154329231360,10022762396642181120,155909637281100595200

%N a(n) = (3*0^n + 4^n*binomial(2*n,n))/4.

%C (1 + (k-1)*sqrt(1-4*k*x))/(k*sqrt(1-4*k*x)) is the g.f. for ((k-1)*0^n + k^n*binomial(2*n,n))/k.

%H Vincenzo Librandi, <a href="/A099045/b099045.txt">Table of n, a(n) for n = 0..800</a>

%F G.f.: (1+3*sqrt(1-16*x))/(4*sqrt(1-16*x)).

%F n*a(n) +8*(-2*n+1)*a(n-1)=0. - _R. J. Mathar_, Nov 24 2012

%F E.g.f.: (3 + exp(8*x) * BesselI(0,8*x)) / 4. - _Ilya Gutkovskiy_, Nov 17 2021

%t Join[{1}, Table[4^(n-1)*Binomial[2*n,n], {n,1,30}]] (* _G. C. Greubel_, Dec 31 2017 *)

%o (Magma) [(3*0^n + 4^n*Binomial(2*n, n))/4: n in [ 0..20]]; // _Vincenzo Librandi_, Nov 24 2012

%o (PARI) for(n=0,30, print1((3*0^n + 4^n*binomial(2*n,n))/4, ", ")) \\ _G. C. Greubel_, Dec 31 2017

%Y Cf. A069723, A088218, A099044, A099046.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Sep 24 2004