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a(n) = (4*n+1)*binomial(3*n,n)/(2*n+1).
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%I #34 Sep 08 2022 08:44:59

%S 1,5,27,156,935,5733,35700,224808,1427679,9126975,58659315,378658800,

%T 2453288292,15944020316,103897691640,678610095504,4441369072335,

%U 29120107628115,191233066114545,1257635016353100

%N a(n) = (4*n+1)*binomial(3*n,n)/(2*n+1).

%C T(2n,n) for A111125. - _Paul Barry_, Apr 19 2007

%C a(n) = A182584(2*n+1). - _Reinhard Zumkeller_, May 06 2012

%H Vincenzo Librandi, <a href="/A052227/b052227.txt">Table of n, a(n) for n = 0..200</a>

%F G.f.: 4*x*F(4/3,5/3;5/2;27*x/4) + 2*sin(1/3*arcsin((3*sqrt(3*x))/2))/sqrt(3*x), where F(a;b;z) is a hypergeometric series. - _Emanuele Munarini_, Jun 06 2011

%F G.f.: (g+1)/((3*g-1)*(g-1)) where g*(1-g)^2 = x. - _Mark van Hoeij_, Nov 10 2011

%F Conjecture: 8*n*(2*n+1)*a(n) +6*(-8*n^2-25*n+13)*a(n-1) -45*(3*n-4)*(3*n-5)*a(n-2)=0. - _R. J. Mathar_, Nov 24 2012

%F a(n) = binomial(3*n+1, n) + binomial(3*n, n-1) for n>=0. - _Paul D. Hanna_, Jul 22 2013

%F G.f.: G(x)*(2*G(x) - 1) / (3 - 2*G(x)), where G(x) = 1 + x*G(x)^3 is the g.f. of A001764. - _Paul D. Hanna_, Jul 22 2013

%F a(n) is the coefficient of [x^n] in (1+x)/(1-x)^(2n+2) and forms the main diagonal in the following table of coefficients:

%F (1+x)/(1-x)^2: [1, 3, 5, 7, 9, 11, 13, 15, 17, ...];

%F (1+x)/(1-x)^4: [1, 5, 14, 30, 55, 91, 140, 204, 285, ...];

%F (1+x)/(1-x)^6: [1, 7, 27, 77, 182, 378, 714, 1254, ...];

%F (1+x)/(1-x)^8: [1, 9, 44, 156, 450, 1122, 2508, 5148, ...];

%F (1+x)/(1-x)^10:[1, 11, 65, 275, 935, 2717, 7007, 16445, ...];

%F (1+x)/(1-x)^12:[1, 13, 90, 442, 1729, 5733, 16744, 44200, ...];

%F (1+x)/(1-x)^14:[1, 15, 119, 665, 2940, 10948, 35700, 104652, ...];

%F (1+x)/(1-x)^16:[1, 17, 152, 952, 4692, 19380, 69768, 224808, ...]; ... - _Paul D. Hanna_, Jul 22 2013

%t Table[(4n + 1)Binomial[3n, n]/(2n + 1), {n, 0, 30}] (* _Harvey P. Dale_, Jan 31 2011 *)

%o (Maxima) makelist(binomial(3*n,n)*(4*n+1)/(2*n+1),n,0,100); /* _Emanuele Munarini_, Jun 06 2011 */

%o (Magma) [(4*n+1)*Binomial(3*n,n)/(2*n+1) : n in [0..30]]; // _Vincenzo Librandi_, Nov 13 2011

%o (Haskell)

%o a052227 n = (a016813 n) * (a005809 n) `div` (a005408 n)

%o -- _Reinhard Zumkeller_, May 06 2012

%o (PARI) {a(n)=binomial(3*n+1, n)+binomial(3*n, n-1)} /* _Paul D. Hanna_, Jul 22 2013 */

%Y Cf. A045721, A016813, A005809, A005408, A227726.

%K easy,nonn

%O 0,2

%A _Barry E. Williams_, Jan 29 2000

%E More terms from _Harvey P. Dale_, Jan 31 2011