A006468 Number of rooted planar maps with 4 faces and n vertices and no isthmuses. (Formerly M3984)

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

1,1

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).

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

Mathematica

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

Prog

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

Crossrefs

Column k=4 of A342981.

Sequence in context: A263783 A023289 A347321 * A144954 A305861 A146195

Adjacent sequences: A006465 A006466 A006467 * A006469 A006470 A006471

Keyword

nonn

Author

N. J. A. Sloane

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

Status

approved