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A003408 a(n) = binomial(3n+6, n).
(Formerly M4643)
9
1, 9, 66, 455, 3060, 20349, 134596, 888030, 5852925, 38567100, 254186856, 1676056044, 11058116888, 73006209045, 482320623240, 3188675231420, 21094923659355, 139646485582065, 925029565741050, 6131164307078475, 40661170824914640, 269807672771096460 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

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

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

EXAMPLE

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.

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

MATHEMATICA

Table[Binomial[3*n + 6, n], {n, 0, 20}] (* Arkadiusz Wesolowski, Apr 02 2012 *)

CROSSREFS

Sequence in context: A051375 A081902 A002695 * A037698 A037607 A055148

Adjacent sequences:  A003405 A003406 A003407 * A003409 A003410 A003411

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Simon Plouffe

EXTENSIONS

Formula found by Simon Plouffe

More terms from James A. Sellers, Aug 21 2000

STATUS

approved

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Last modified June 23 17:11 EDT 2021. Contains 345402 sequences. (Running on oeis4.)