A000184 Number of genus 0 rooted maps with 3 faces with n vertices.

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

Author

N. J. A. Sloane

Keywords

• nonn

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

Crossrefs

Column 3 of A269920.

Column 0 of A270407.

Cf. A000108, A029887.

Programs

Magma

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

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 *)

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

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)

Extensions

More terms from Sean A. Irvine, Nov 14 2010