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A006343 Arkons: number of elementary maps with n-1 nodes.
(Formerly M3400)
13

%I M3400 #51 Feb 20 2020 03:38:30

%S 1,0,1,1,4,10,34,112,398,1443,5387,20482,79177,310102,1228187,4910413,

%T 19792582,80343445,328159601,1347699906,5561774999,23052871229,

%U 95926831442,400587408251,1678251696379,7051768702245,29710764875014

%N Arkons: number of elementary maps with n-1 nodes.

%D K. Appel and W. Haken, Every planar map is four colorable. With the collaboration of J. Koch. Contemporary Mathematics, 98. American Mathematical Society, Providence, RI, 1989. xvi+741 pp. ISBN: 0-8218-5103-9.

%D F. R. Bernhart, Topics in Graph Theory Related to the Five Color Conjecture. Ph.D. Dissertation, Kansas State Univ., 1974.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Reinhard Zumkeller, <a href="/A006343/b006343.txt">Table of n, a(n) for n = 0..1000</a>

%H F. R. Bernhart, <a href="http://dx.doi.org/10.1016/S0012-365X(99)00054-0">Catalan, Motzkin and Riordan numbers</a>, Discr. Math., 204 (1999), 73-112.

%H F. R. Bernhart, <a href="/A000296/a000296_1.pdf">Fundamental chromatic numbers</a>, Unpublished. (Annotated scanned copy)

%H F. R. Bernhart & N. J. A. Sloane, <a href="/A001683/a001683.pdf">Correspondence, 1977</a>

%H F. R. Bernhart & N. J. A. Sloane, <a href="/A006343/a006343.pdf">Emails, April-May 1994</a>

%H G. D. Birkhoff and D. C. Lewis, <a href="http://dx.doi.org/10.1090/S0002-9947-1946-0018401-4">Chromatic polynomials</a>, Trans. Amer. Math. Soc. 60, (1946). 355-451.

%F a(n-1) = Sum (n-k-1)^(-1)*binomial(n, k)*binomial(2*n-3*k-4, n-2*k-2); k = 0..[ (n-2)/2 ], n >= 3.

%F From _Peter Bala_, Jun 22 2015: (Start)

%F O.g.f. A(x) equals 1/x * series reversion ( x/(1 + x^2*C(x) ), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. for A000108.

%F A(x) is an algebraic function satisfying x^3*A^3(x) - (x - 1)*A^2(x) + (x - 2)*A(x) + 1 = 0. (End)

%F a(n) ~ sqrt(s*(1 - s + 3*r^2*s^2) / (1 - r + 3*r^3*s)) / (2*sqrt(Pi) * n^(3/2) * r^(n - 1/2)), where r = 0.2229935155751761877673240243525445951244491757706... and s = 1.116796494086474135831052534637944909439048671327... are real roots of the system of equations 1 + (r-2)*s + r^3*s^3 = (r-1)*s^2, r + 2*s + 3*r^3*s^2 = 2 + 2*r*s. - _Vaclav Kotesovec_, Nov 27 2017

%F Conjecture: D-finite with recurrence: -(n+3)*(n-1)*a(n) +(11*n^2-2*n-45)*a(n-1) -(37*n+29)*(n-3)*a(n-2) +(29*n^2-125*n+78)*a(n-3) +(61*n-106)*(n-3)*a(n-4) -155*(n-3)*(n-4)*a(n-5)=0. - _R. J. Mathar_, Feb 20 2020

%p A006343:=n->add(binomial(n,k)*binomial(2*n-3*k-4,n-2*k-2)/(n-k-1), k=0..(n-2)/2): (1, seq(A006343(n), n=1..30)); # _Wesley Ivan Hurt_, Jun 22 2015

%t a[n_] := Sum[ Binomial[n, k]*Binomial[2*n-3*k-4, n-2*k-2]/(n-k-1), {k, 0, (n-2)/2}]; a[0] = 1; Table[a[n], {n, 0, 26}] (* _Jean-François Alcover_, Dec 14 2012, from formula *)

%o (Haskell)

%o a006343 0 = 1

%o a006343 n = sum $ zipWith div

%o (zipWith (*) (map (a007318 n) ks)

%o (map (\k -> a007318 (2*n - 3*k - 4) (n - 2*k - 2)) ks))

%o (map (toInteger . (n - 1 -)) ks)

%o where ks = [0 .. (n - 2) `div` 2]

%o -- _Reinhard Zumkeller_, Aug 23 2012

%Y Cf. A000108, A000934, A007318.

%K easy,nonn,nice

%O 0,5

%A _N. J. A. Sloane_

%E Erroneously duplicated term 4 removed per Frank Bernhart's report by _Max Alekseyev_, Feb 11 2010

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Last modified March 28 08:22 EDT 2024. Contains 371236 sequences. (Running on oeis4.)