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

1, 0, 1, 1, 4, 10, 34, 112, 398, 1443, 5387, 20482, 79177, 310102, 1228187, 4910413, 19792582, 80343445, 328159601, 1347699906, 5561774999, 23052871229, 95926831442, 400587408251, 1678251696379, 7051768702245, 29710764875014

Offset

0,5

References

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.

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

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

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 (1999), 73-112.

F. R. Bernhart, Fundamental chromatic numbers, Unpublished. (Annotated scanned copy)

F. R. Bernhart & N. J. A. Sloane, Correspondence, 1977

F. R. Bernhart & N. J. A. Sloane, Emails, April-May 1994

G. D. Birkhoff and D. C. Lewis, Chromatic polynomials, Trans. Amer. Math. Soc. 60, (1946). 355-451.

Formula

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.

From Peter Bala, Jun 22 2015: (Start)

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.

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

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

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

Maple

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

Mathematica

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 *)

Prog

(Haskell)
a006343 0 = 1
a006343 n = sum $ zipWith div
   (zipWith (*) (map (a007318 n) ks)
                (map (\k -> a007318 (2*n - 3*k - 4) (n - 2*k - 2)) ks))
   (map (toInteger . (n - 1 -)) ks)
   where ks = [0 .. (n - 2) `div` 2]
-- Reinhard Zumkeller, Aug 23 2012

Crossrefs

Cf. A000108, A000934, A007318.

Sequence in context: A274479 A231524 A182645 * A149173 A149174 A283070

Adjacent sequences: A006340 A006341 A006342 * A006344 A006345 A006346

Keyword

easy,nonn,nice

Author

N. J. A. Sloane

Extensions

Erroneously duplicated term 4 removed per Frank Bernhart's report by Max Alekseyev, Feb 11 2010

Status

approved