OFFSET
0,2
COMMENTS
T(2n,n) for A111125. - Paul Barry, Apr 19 2007
a(n) = A182584(2*n+1). - Reinhard Zumkeller, May 06 2012
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
FORMULA
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
G.f.: (g+1)/((3*g-1)*(g-1)) where g*(1-g)^2 = x. - Mark van Hoeij, Nov 10 2011
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
a(n) = binomial(3*n+1, n) + binomial(3*n, n-1) for n>=0. - Paul D. Hanna, Jul 22 2013
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
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:
(1+x)/(1-x)^2: [1, 3, 5, 7, 9, 11, 13, 15, 17, ...];
(1+x)/(1-x)^4: [1, 5, 14, 30, 55, 91, 140, 204, 285, ...];
(1+x)/(1-x)^6: [1, 7, 27, 77, 182, 378, 714, 1254, ...];
(1+x)/(1-x)^8: [1, 9, 44, 156, 450, 1122, 2508, 5148, ...];
(1+x)/(1-x)^10:[1, 11, 65, 275, 935, 2717, 7007, 16445, ...];
(1+x)/(1-x)^12:[1, 13, 90, 442, 1729, 5733, 16744, 44200, ...];
(1+x)/(1-x)^14:[1, 15, 119, 665, 2940, 10948, 35700, 104652, ...];
(1+x)/(1-x)^16:[1, 17, 152, 952, 4692, 19380, 69768, 224808, ...]; ... - Paul D. Hanna, Jul 22 2013
MATHEMATICA
Table[(4n + 1)Binomial[3n, n]/(2n + 1), {n, 0, 30}] (* Harvey P. Dale, Jan 31 2011 *)
PROG
(Maxima) makelist(binomial(3*n, n)*(4*n+1)/(2*n+1), n, 0, 100); /* Emanuele Munarini, Jun 06 2011 */
(Magma) [(4*n+1)*Binomial(3*n, n)/(2*n+1) : n in [0..30]]; // Vincenzo Librandi, Nov 13 2011
(Haskell)
a052227 n = (a016813 n) * (a005809 n) `div` (a005408 n)
-- Reinhard Zumkeller, May 06 2012
(PARI) {a(n)=binomial(3*n+1, n)+binomial(3*n, n-1)} /* Paul D. Hanna, Jul 22 2013 */
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Barry E. Williams, Jan 29 2000
EXTENSIONS
More terms from Harvey P. Dale, Jan 31 2011
STATUS
approved