A006468 Number of rooted planar maps with 4 faces and n vertices and no isthmuses.

5, 37, 150, 449, 1113, 2422, 4788, 8790, 15213, 25091, 39754, 60879, 90545, 131292, 186184, 258876, 353685, 475665, 630686, 825517, 1067913, 1366706, 1731900, 2174770, 2707965, 3345615, 4103442, 4998875, 6051169, 7281528, 8713232, 10371768, 12284965, 14483133, 16999206

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

References

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

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1000
• Simon Plouffe, Approximations of generating functions and a few conjectures, Master's thesis, UQAM, 1992; arXiv:0911.4975 [math.NT], 2009.
• T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus. III: Nonseparable maps, J. Combinatorial Theory Ser. B 18 (1975), 222-259, Table VIb, f=5.
• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Crossrefs

Column k=4 of A342981.

Programs

Mathematica

A006468[n_] := n*(n + 1)*(n + 2)*(n*(n*(2*n + 33) + 142) + 123)/360;
Array[A006468, 50] (* Paolo Xausa, Aug 20 2025 *)

PARI

a(n) = {n *(n+1) *(n+2) *(2*n^3 + 33*n^2 + 142*n + 123) /360} \\ Andrew Howroyd, Apr 02 2021

Formula

G.f.: -x*(x^3-4*x^2+2*x+5)/(x-1)^7, equivalent to a(n) = n *(n+1) *(n+2) *(2*n^3 +33*n^2 +142*n +123) /360, conjectured in Simon Plouffe's Master's thesis, 1992.

The above conjecture is true. - Andrew Howroyd, Apr 02 2021

Extensions

Title improved and a(13)-a(14) from Sean A. Irvine, Apr 24 2017

Terms a(15) and beyond from Andrew Howroyd, Apr 02 2021