%I #36 Oct 04 2024 12:25:18

%S 1,9,90,945,10206,112266,1250964,14073345,159497910,1818276174,

%T 20827527084,239516561466,2763652632300,31979409030900,

%U 370961144758440,4312423307816865,50227047938102310,585982225944526950,6846739692614999100,80106854403595489470,938394580156404305220

%N a(n) = 3^n*binomial(2*n+1, n).

%H Vincenzo Librandi, <a href="/A098399/b098399.txt">Table of n, a(n) for n = 0..900</a>

%F G.f.: (1-sqrt(1-12*x))/(6*x*sqrt(1-12*x)).

%F E.g.f.: a(n) = n!* [x^n] exp(6*x)*(BesselI(0, 6*x) + BesselI(1, 6*x)). - _Peter Luschny_, Aug 25 2012

%F (n+1)*a(n) - 6*(2*n+1)*a(n-1) = 0. - _R. J. Mathar_, Nov 24 2012

%F From _G. C. Greubel_, Dec 27 2023: (Start)

%F a(n) = 3^n * (2*n+1)*A000108(n).

%F a(n) = (2*n+1)*A005159(n).

%F a(n) = 3^n * A001700(n). (End)

%F From _Amiram Eldar_, Jan 16 2024: (Start)

%F Sum_{n>=0} 1/a(n) = 6/11 + 72*arcsin(1/(2*sqrt(3)))/(11*sqrt(11)).

%F Sum_{n>=0} (-1)^n/a(n) = 6/13 + 72*arcsinh(1/(2*sqrt(3)))/(13*sqrt(13)). (End)

%p Z:=(1-sqrt(1-3*z))*4^n/sqrt(1-3*z)/6: Zser:=series(Z, z=0, 32): seq(coeff(Zser, z, n), n=1..18); # _Zerinvary Lajos_, Jan 01 2007

%t Table[3^n Binomial[2n+1,n], {n,0,20}] (* _Harvey P. Dale_, Mar 28 2012 *)

%o (Magma) [3^n*Binomial(2*n+1, n): n in [ 0..20]]; // _Vincenzo Librandi_, Nov 24 2012

%o (PARI) a(n)=binomial(2*n+1,n)*3^n \\ _Charles R Greathouse IV_, Oct 23 2023

%o (SageMath) [3^n*binomial(2*n+1, n) for n in range(21)] # _G. C. Greubel_, Dec 27 2023

%Y Cf. A000108, A001700, A005159, A069720, A069723, A098400.

%K easy,nonn,changed

%O 0,2

%A _Paul Barry_, Sep 06 2004