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A046646 a(n) = 2*binomial(3*n-3, n-1)/(2*n-1) for n >= 2, and a(1) = 1.

%I A046646 #39 Feb 23 2024 06:06:13
%S A046646 1,2,6,24,110,546,2856,15504,86526,493350,2861430,16829280,100134216,
%T A046646 601661144,3645533040,22249511328,136657509918,844061090670,
%U A046646 5239262085330,32665844580600,204480219795390,1284624902435490
%N A046646 a(n) = 2*binomial(3*n-3, n-1)/(2*n-1) for n >= 2, and a(1) = 1.
%C A046646 Number of certain rooted planar maps.
%H A046646 Vincenzo Librandi, <a href="/A046646/b046646.txt">Table of n, a(n) for n = 1..200</a>
%H A046646 Daniel Birmajer, Juan B. Gil, Michael D. Weiner, <a href="https://arxiv.org/abs/1707.09918">Bounce statistics for rational lattice paths</a>, arXiv:1707.09918 [math.CO], 2017, p. 10.
%H A046646 W. G. Brown, <a href="http://dx.doi.org/10.4153/CJM-1963-056-7">Enumeration of non-separable planar maps</a>, Canad. J. Math., 15 (1963), 526-545.
%H A046646 W. G. Brown, <a href="/A000087/a000087.pdf">Enumeration of non-separable planar maps</a> [Annotated scanned copy]
%F A046646 From _Emeric Deutsch_, Mar 03 2004: (Start)
%F A046646 a(n) = 2*binomial(3*n-3, n-1)/(2*n-1) for n >= 2, and a(1) = 1.
%F A046646 a(n) = 2*A001764(n-1) for n >= 2. (End)
%F A046646 a(n) = (n+1) * A000139(n). - _F. Chapoton_, Feb 23 2024
%F A046646 G.f.: (1+g)/(1-g) where g*(1-g)^2 = x. - _Mark van Hoeij_, Nov 10 2011
%p A046646 alias(PS=ListTools:-PartialSums): A046646List := proc(m) local A, P, n;
%p A046646 A := [1,2]; P := [2]; for n from 1 to m - 2 do P := PS(PS([op(P), P[-1]]));
%p A046646 A := [op(A), P[-1]] od; A end: A046646List(22); # _Peter Luschny_, Mar 26 2022
%t A046646 Join[{1},Table[(2*Binomial[3n-3,n-1])/(2n-1),{n,2,30}]] (* _Harvey P. Dale_, Oct 12 2013 *)
%o A046646 (Magma) [1] cat [2*Binomial(3*n-3,n-1)/(2*n-1): n in [2..30]]; // _Vincenzo Librandi_, Oct 13 2013
%Y A046646 A diagonal of A046651.
%Y A046646 Cf. A046647, A005470, A000139, A001764.
%K A046646 nonn,easy
%O A046646 1,2
%A A046646 _N. J. A. Sloane_
%E A046646 More terms from _Emeric Deutsch_, Mar 03 2004
%E A046646 New name using a formula of _Emeric Deutsch_ by _Peter Luschny_, Feb 23 2024