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A002829
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Number of trivalent (or cubic) labeled graphs with 2n nodes.
(Formerly M5346 N2324)
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10
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1, 0, 1, 70, 19355, 11180820, 11555272575, 19506631814670, 50262958713792825, 187747837889699887800, 976273961160363172131825, 6840300875426184026353242750, 62870315446244013091262178375075
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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REFERENCES
| L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 279.
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983.
Goulden, I. P.; Jackson, D. M.; Reilly, J. W.; The Hammond series of a symmetric function and its application to $P$-recursiveness. SIAM J. Algebraic Discrete Methods 4 (1983), no. 2, 179-193.
R. C. Read, Some Enumeration Problems in Graph Theory. Ph.D. Dissertation, Department of Mathematics, Univ. London, 1958.
R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1977.
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).
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LINKS
| R. W. Robinson, Table of n, a(n) for n = 0..30
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FORMULA
| E.g.f. f(x) = Sum_{n >= 0} a(2 * n) * x^n/(2 * n)! satisfies differential equation 6 * x^2 * ( - x^2 - 2 * x + 2) * diff(f(x), x, x) - (x^5 + 6 * x^4 + 6 * x^3 - 32 * x + 8) * diff(f(x), x) + x/6 * ( - x^2 - 2 * x + 2)^2 * f(x) = 0.
Recurrence: a(2 * n) = (2 * n)!/n! * v(n) where 48 * v(n) + ( - 72 * n^2 + 24 * n + 48) * v(n - 1) + (72 * n^3 - 432 * n^2 + 788 * n - 428) * v(n - 2) + (36 * n^4 - 324 * n^3 + 1052 * n^2 - 1428 * n + 664) * v(n - 3) + (36 * n^4 - 360 * n^3 + 1260 * n^2 - 1800 * n + 864) * v(n - 4) + (6 * n^5 - 94 * n^4 + 550 * n^3 - 1490 * n^2 + 1844 * n - 816) * v(n - 5) + ( - n^5 + 15 * n^4 - 85 * n^3 + 225 * n^2 - 274 * n + 120) * v(n - 6) = 0.
$\dsum\limits_{a_{2}=0}^{2n}\dsum\limits_{c=0}^{\min \{\lfloor (3n-a_{2})/3\rfloor ,\lfloor (2n-a_{2})/2\rfloor \}}\dsum\limits_{b=0}^{\min \{\lfloor (3n-a_{2}-3c)/2\rfloor ,\lfloor (2n-a_{2}-2c)/2\rfloor \}}\frac{% (-1)^{a_{2}+b}(2n)!(2\allowbreak (3n-a_{2}-2b-3c))!}{2^{(\allowbreak 5n-a_{2}-2b-4c)}3^{(2n-a_{2}-2b-c)}\allowbreak (3n-a_{2}-2b-3c)!a_{2}!b!c!(2n-a_{2}-2b-2c)!}$ [From Shanzhen Gao (sgao2(AT)fau.edu), Jun 05 2009]
E.g.f.: hypergeom([1/6, 5/6],[],12*x/(x^2+8*x+4)^(3/2))*exp(-ln(1/4*x^2+2*x+1)/4 - x/3 + (x^2+8*x+4)^(3/2)/(24*x) - 1/(3*x) - x^2/24 - 1) - Multiply x^i by (2*i)! to get the generating function. - Mark van Hoeij, Nov 07 2011
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MAPLE
| Contribution from R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 31 2010: (Start)
A002829aux := proc(i) local a, j, k ; a := 0 ; for j from 0 to i do for k from 0 to 2*(i-j) do a := a+(-1)^(j+k)/j!*doublefactorial(2*i+2*k-1)/3^k/k!/(2*i-2*j-k)! ; end do: end do: a*3^i/2^i ; end proc:
A002829 := proc(n) (2*n)!/6^n*add( A002829aux(i)/(n-i)!, i=0..n) ; end proc: seq(A002829(n), n=0..6) ; (End)
egf := hypergeom([1/6, 5/6], [], 12*x/(x^2+8*x+4)^(3/2)) * exp(-ln(1/4*x^2+2*x+1)/4 - x/3 + (x^2+8*x+4)^(3/2)/(24*x) - 1/(3*x) - x^2/24 - 1):
ser := convert(series(egf, x=0, 30), polynom):
seq(coeff(ser, x, i) * (2*i)!, i=0..degree(ser)); - Mark van Hoeij, Nov 07 2011
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CROSSREFS
| A diagonal of A059441. Cf. A005814.
Sequence in context: A103157 A007099 A004109 * A177637 A145410 A177638
Adjacent sequences: A002826 A002827 A002828 * A002830 A002831 A002832
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms and formula from Vladeta Jovovic (vladeta(AT)eunet.rs), Mar 25 2001
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