A144485 - OEIS

%I #22 Dec 07 2024 05:45:27

%S 2,20,168,1320,10010,74256,542640,3922512,28120950,200300100,

%T 1419269280,10013421600,70394353848,493362138080,3448674255840,

%U 24051721745568,167405449649550,1163116182943260,8068463611408200,55891260077406600

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

%H Vincenzo Librandi, <a href="/A144485/b144485.txt">Table of n, a(n) for n = 0..200</a>

%H Ömer Eğecioğlu, Timothy Redmond, and Charles Ryavec, <a href="https://www.emis.de/journals/EJC/Volume_15/Abstracts/v15i1r6.html">Almost product evaluation of Hankel Determinants</a>, The Electronic Journal of Combinatorics, Vol. 15, No. 1 (2008), #R6; <a href="https://arxiv.org/abs/0704.3398">arXiv preprint</a>, arXiv:0704.3398 [math.CO], 2007.

%F a(n) = (3n+2)*A045721(n). - _R. J. Mathar_, Feb 01 2014

%F a(n) = 2*A090763(n). - _Alois P. Heinz_, Feb 01 2014

%F From _Amiram Eldar_, Dec 07 2024: (Start)

%F a(n) = 2 * (n+1) * A005809(n+1) / 3.

%F Sum_{n>=0} 1/a(n) = (3/2) * A210453. (End)

%p a:= proc(n) option remember; `if`(n=0, 2,

%p       3*(3*n+1)*(3*n+2)*a(n-1)/(2*n*(2*n+1)))

%p     end:

%p seq(a(n), n=0..30);  # _Alois P. Heinz_, Feb 01 2014

%t a[k_] = (3k + 2)Binomial[3k + 1, k]; Table[a[k], {k, 0, 30}]

%o (Magma) [(3*n+2)*Binomial(3*n+1, n): n in [0..20]]; // _Vincenzo Librandi_, Feb 14 2014

%Y Cf. A005809, A045721, A090763, A210453.

%K nonn,easy

%O 0,1

%A _Roger L. Bagula_, Oct 12 2008