A000184 Number of genus 0 rooted maps with 3 faces with n vertices. (Formerly M2128 N0843)

2, 22, 164, 1030, 5868, 31388, 160648, 795846, 3845020, 18211380, 84876152, 390331292, 1775032504, 7995075960, 35715205136, 158401506118, 698102372988, 3059470021316, 13341467466520, 57918065919924, 250419305769512, 1078769490401032, 4631680461623664, 19825379450255900, 84622558822506328, 360270317908904328, 1530148541536781488, 6484511936352543096, 27423786092731382000, 115756362341775227888

Offset

2,1

References

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).

T. R. S. Walsh, Combinatorial Enumeration of Non-Planar Maps. Ph.D. Dissertation, Univ. of Toronto, 1971.

Links

Alois P. Heinz, Table of n, a(n) for n = 2..500

Richard P. Stanley, CATALAN ADDENDUM, version of Jul 19, 2008, p. 24. [From Jonathan Vos Post, Aug 16 2008]

M. S. Tokmachev, Correlations Between Elements and Sequences in a Numerical Prism, Bulletin of the South Ural State University, Ser. Mathematics. Mechanics. Physics, 2019, Vol. 11, No. 1, 24-33.

W. T. Tutte, On the enumeration of planar maps, Bull. Amer. Math. Soc. 74 1968 64-74.

T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus, J. Comb. Thy B13 (1972), 122-141 and 192-218.

Notes

Formula

a(n) = 2 * A029887(n-2). - Ralf Stephan, Aug 17 2004

a(n) = 4^n*Gamma(n+3/2)/(3*sqrt(Pi)*Gamma(n)) - n*4^(n-1). - Mark van Hoeij, Jul 06 2010

From G. C. Greubel, Jul 18 2024: (Start)

a(n) = (n/12)*( (n+1)*(n+2)*Catalan(n+1) - 3*4^n ).

G.f.: x*(1 - sqrt(1 - 4*x))/(1-4*x)^(5/2).

E.g.f.: (x/3)*exp(2*x)*( - 3*exp(2*x) + 3*(1+2*x)*BesselI(0, 2*x) + (3+8*x)*BesselI(1, 2*x) + 2*x*BesselI(2, 2*x) ). (End)

Mathematica

a[n_] := 1/12*(2^(n+1)*(2*n+1)!!/(n-1)!-3*4^n*n); Table[a[n], {n, 2, 31}] (* Jean-François Alcover, Mar 12 2014 *)

Prog

(Magma)
[n*((n+1)*(n+2)*Catalan(n+1) - 3*4^n)/12: n in [2..30]]; // G. C. Greubel, Jul 18 2024
(SageMath)
[n*(2*(2*n+1)*binomial(2*n, n) - 3*4^n)//12 for n in range(2, 30)] # G. C. Greubel, Jul 18 2024

Crossrefs

Column 3 of A269920.

Column 0 of A270407.

Cf. A000108, A029887.

Sequence in context: A279380 A230835 A270407 * A007613 A346796 A279801

Adjacent sequences: A000181 A000182 A000183 * A000185 A000186 A000187

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Sean A. Irvine, Nov 14 2010

Status

approved