A006417 Number of loopless rooted planar maps with 4 faces and n vertices and no isthmuses.

1, 20, 131, 469, 1262, 2862, 5780, 10725, 18647, 30784, 48713, 74405, 110284, 159290, 224946, 311429, 423645, 567308, 749023, 976373, 1258010, 1603750, 2024672, 2533221, 3143315, 3870456, 4731845, 5746501, 6935384, 8321522, 9930142, 11788805, 13927545, 16379012

Offset: 2

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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 = 2..1000
• 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.
• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Crossrefs

Column k=4 of A342980.

Programs

PARI

a(n)={if(n<2, 0, (2*n^6 + 39*n^5 + 125*n^4 - 345*n^3 - 1027*n^2 + 846*n + 2160)/360)} \\ Andrew Howroyd, Apr 01 2021

Formula

From Colin Barker, Apr 08 2013: (Start)

a(n) = (2160+846*n-1027*n^2-345*n^3+125*n^4+39*n^5+2*n^6)/360.

G.f.: -x^2*(5*x^6-29*x^5+65*x^4-63*x^3+12*x^2+13*x+1) / (x-1)^7. (End)

E.g.f.: exp(x)*(2160 - 360*x - 540*x^2 + 1560*x^3 + 645*x^4 + 69*x^5 + 2*x^6)/360 - 6 - 5*x. - Stefano Spezia, Jul 18 2024

Extensions

Title improved by Sean A. Irvine, Apr 03 2017

Terms a(14) and beyond from Andrew Howroyd, Apr 01 2021