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A000287
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Number of rooted polyhedral graphs with n edges.
(Formerly M3290 N1326)
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7
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1, 0, 4, 6, 24, 66, 214, 676, 2209, 7296, 24460, 82926, 284068, 981882, 3421318, 12007554, 42416488, 150718770, 538421590, 1932856590, 6969847486, 25237057110, 91729488354, 334589415276, 1224445617889, 4494622119424
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OFFSET
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6,3
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COMMENTS
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a(n) appears to be odd if and only if n = 2^k - 2 for some integer k >= 3. - Lewis Chen, May 05 2019
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REFERENCES
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Handbook of Combinatorics, North-Holland '95, p. 892. (Gives different last term)
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Tutte, W. T. Three-connected planar maps. Proceedings of the Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1971), pp. 43--52. Dept. Comput. Sci., Univ. Manitoba, Winnipeg, Man., 1971. MR0335323 (49 #105). - From N. J. A. Sloane, Jun 05 2012
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LINKS
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FORMULA
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a(n) = b(n-1) + 2*(-1)^n, n >= 4, where b(3)=2, b(n) = (2*(2*n)!/(n!)^2 - (27*n^2+9*n-2)*b(n-1)) / (54*n^2-90*n+32). - Sean A. Irvine, Apr 14 2010
(n+4)*a(n) = ((3/2)*n - 3)*a(n-1) + (8*n + 4)*a(n-2) + ((15/2)*n + 6)*a(n-3) + (2*n + 3)*a(n-4). - Simon Plouffe, Feb 09 2012
G.f.: x^2 - 2*x^3/(1+x) + x*(2*x^2-10*x-1+(1-4*x)^(3/2))/(2*(x+2)^3).
0 = x*(x+1)^2*(x+2)*(4*x-1)*y' + 2*(x^2-11*x+1)*(x+1)^2*y + 10*x^6, where y is the g.f. (End)
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EXAMPLE
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G.f. = x^6 + 4*x^8 + 6*x^9 + 24*x^10 + 66*x^11 + 214*x^12 + 676*x^13 + ...
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MATHEMATICA
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a[6] = 1; a[n_] := a[n] = ((9*(5 - 3*n)*n - 16)*a[n-1]*((n-1)!)^2 + 2*((-1)^n*(9*n*(3*n - 17) + 160)*((n-1)!)^2 + ((2*n - 2)!)))/(2*(9*n*(3*n - 11) + 88)*((n-1)!)^2); Table[ a[n], {n, 6, 31}] (* Jean-François Alcover, Oct 04 2011, after formula *)
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PROG
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(PARI)
seq(N) = {
my(x='x+O('x^(N+5)));
Vec(x^2 - 2*x^3/(1+x) + x*(2*x^2-10*x-1+(1-4*x)^(3/2))/(2*(x+2)^3));
};
seq(26)
\\ test: y=Ser(seq(101))*x^6; 0 == x*(x+1)^2*(x+2)*(4*x-1)*y' + 2*(x^2-11*x+1)*(x+1)^2*y + 10*x^6
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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