|
%I #27 Feb 23 2024 06:06:40
%S 2,3,12,60,330,1911,11424,69768,432630,2713425,17168580,109390320,
%T 700939512,4512458580,29164264320,189120846288,1229917589262,
%U 8018580361365,52392620853300,342991368096300,2249282417749290
%N a(n) = (n + 2) * binomial(3*n, n) / (2*n + 1).
%C A second-order recursive sequence.
%H L. Carlitz, <a href="http://www.fq.math.ca/Scanned/11-2/carlitz.pdf">Enumeration of two-line arrays</a>, Fib. Quart., 11 (1973), pp. 113-130.
%F a(n) = (n+2)*c(2; n), where c(2; n) = binomial(3*n, n)/(2*n+1) (A001764).
%F c(2; n) is equivalent to Eq. (6.22) on p. 129 of the Carlitz reference.
%F a(n) = binomial(n+2, 2) * A0000139(n). - _F. Chapoton_, Feb 23 2024
%F G.f.: (2-5*g)/((3*g-1)*(g-1)) where g*(1-g)^2 = x. - _Mark van Hoeij_, Nov 10 2011
%F Conjecture: 2*n*(2*n+1)*a(n) + (-47*n^2+50*n-12)*a(n-1) + 15*(3*n-4)*(3*n-5)*a(n-2) = 0. - _R. J. Mathar_, Sep 27 2012
%t Table[(n+2) Binomial[3n,n]/(2n+1),{n,0,20}] (* _Harvey P. Dale_, Mar 23 2013 *)
%Y Cf. A001764, A000108, A000139.
%K easy,nonn
%O 0,1
%A _Barry E. Williams_, Jan 27 2000
%E More terms from _James A. Sellers_, Jan 31 2000
%E New name using a formula of the author by _Peter Luschny_, Feb 23 2024
|
