A000365 Number of genus 0 rooted planar maps with 4 faces and n vertices. (Formerly M4022 N1669)

5, 93, 1030, 8885, 65954, 442610, 2762412, 16322085, 92400330, 505403910, 2687477780, 13957496098, 71053094420, 355548314180, 1752827693528, 8529176056965, 41026491589722, 195327793313790, 921451498774660, 4311086414580022, 20019238138410940

Offset

3,1

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

T. D. Noe, Table of n, a(n) for n = 3..200

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

G.f.: x^2*(1-sqrt(1-4*x))*(7+4*x-2*sqrt(1-4*x))/(2*(4*x-1)^4). - corrected for right offset by Vaclav Kotesovec, Aug 13 2013

a(n) ~ n^3*4^n/24 * (1-4/(sqrt(Pi*n))). - Vaclav Kotesovec, Aug 13 2013

Mathematica

nn = 20; CoefficientList[Series[x^2 (1 - Sqrt[1 - 4 x]) (7 + 4 x - 2 Sqrt[1 - 4 x])/(2 (4 x - 1)^4), {x, 0, nn}], x] (* T. D. Noe, Jun 19 2012 *)

Prog

(PARI) seq(n)={my(g=sqrt(1-4*x + O(x*x^n))); Vec((1-g)*(7+4*x-2*g)/(2*(1-4*x)^4))} \\ Andrew Howroyd, Mar 27 2021

Crossrefs

Column 4 of A269920.

Column 0 of A270408.

Sequence in context: A152283 A205344 A270408 * A209471 A012784 A136097

Adjacent sequences: A000362 A000363 A000364 * A000366 A000367 A000368

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Sean A. Irvine, Nov 14 2010

Status

approved