|
%I M4643 #59 Mar 02 2024 13:44:21
%S 1,9,66,455,3060,20349,134596,888030,5852925,38567100,254186856,
%T 1676056044,11058116888,73006209045,482320623240,3188675231420,
%U 21094923659355,139646485582065,925029565741050,6131164307078475,40661170824914640,269807672771096460
%N a(n) = binomial(3n+6, n).
%C Number of connected graphs without crossing edges on n+3 nodes on a circle and having exactly 1 interior face. - _Emeric Deutsch_, Nov 06 2001
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A003408/b003408.txt">Table of n, a(n) for n = 0..200</a>
%H C. Domb and A. J. Barrett, <a href="http://dx.doi.org/10.1016/0012-365X(74)90081-8">Enumeration of ladder graphs</a>, Discrete Math. 9 (1974), 341-358.
%H C. Domb & A. J. Barrett, <a href="/A003408/a003408.pdf">Enumeration of ladder graphs</a>, Discrete Math. 9 (1974), 341-358. (Annotated scanned copy)
%H C. Domb & A. J. Barrett, <a href="/A001764/a001764.pdf">Notes on Table 2 in "Enumeration of ladder graphs"</a>, Discrete Math. 9 (1974), 55. (Annotated scanned copy)
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>
%F a(n) = Sum_{k=0..n} binomial(2*n+k+5,k). - _Arkadiusz Wesolowski_, Apr 02 2012
%F 2*n*(n+3)*(2*n+5)*a(n) - 3*(3*n+5)*(3*n+4)*(n+2)*a(n-1) = 0. - _R. J. Mathar_, Feb 05 2013
%F From Karol A. Penson, Feb 28 2024. (Start)
%F O.g.f.(z) = hypergeometric3F2([7/3, 8/3, 3], [7/2, 4], (27*z)/4).
%F O.g.f.(z) = g satisfies the algebraic equation: 1 + (-15*z^2+9*z-1)*g + (27*z-4)*z^3*g^2 + (27*z-4)*z^6*g^3 = 0.
%F a(n) is not a positive definite sequence, i.e. it cannot be represented as the n-th power moment of a positive weight function. (End)
%e a(0)=1 because among the 4 non-crossing connected graphs on 3 nodes on a circle only the triangle has exactly 1 interior face.
%p a:=n->sum(binomial(2*n-2,n+j)*binomial(n-1,n-j),j=0..n): seq(a(n), n=3..22); # _Zerinvary Lajos_, Jan 29 2007
%p R := RootOf(x-t*(t-1)^2,t); ogf := series(1/((1-3*R)*(1-R)^6),x=0,20); # _Mark van Hoeij_, Nov 08 2011
%t Table[Binomial[3*n + 6, n], {n, 0, 20}] (* _Arkadiusz Wesolowski_, Apr 02 2012 *)
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, _Simon Plouffe_
%E Formula found by _Simon Plouffe_
%E More terms from _James A. Sellers_, Aug 21 2000
| 