A003408 a(n) = binomial(3n+6, n).
1, 9, 66, 455, 3060, 20349, 134596, 888030, 5852925, 38567100, 254186856, 1676056044, 11058116888, 73006209045, 482320623240, 3188675231420, 21094923659355, 139646485582065, 925029565741050, 6131164307078475, 40661170824914640, 269807672771096460
Offset: 0
Examples
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.
References
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- T. D. Noe, Table of n, a(n) for n = 0..200
- C. Domb and A. J. Barrett, Enumeration of ladder graphs, Discrete Math. 9 (1974), 341-358.
- C. Domb & A. J. Barrett, Enumeration of ladder graphs, Discrete Math. 9 (1974), 341-358. (Annotated scanned copy)
- C. Domb & A. J. Barrett, Notes on Table 2 in "Enumeration of ladder graphs", Discrete Math. 9 (1974), 55. (Annotated scanned copy)
- Milan Janjic, Two Enumerative Functions
Programs
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Maple
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 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
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Mathematica
Table[Binomial[3*n + 6, n], {n, 0, 20}] (* Arkadiusz Wesolowski, Apr 02 2012 *)
Formula
a(n) = Sum_{k=0..n} binomial(2*n+k+5,k). - Arkadiusz Wesolowski, Apr 02 2012
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
From Karol A. Penson, Feb 28 2024. (Start)
O.g.f.(z) = hypergeometric3F2([7/3, 8/3, 3], [7/2, 4], (27*z)/4).
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.
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)
Extensions
Formula found by Simon Plouffe
More terms from James Sellers, Aug 21 2000
Comments