A006429 Number of loopless tree-rooted planar maps with 4 vertices and n faces.

0, 4, 135, 1368, 7350, 28400, 89073, 241220, 585057, 1301420, 2699125, 5282172, 9842430, 17584416, 30289835, 50530680, 81940901, 129557940, 200246795, 303220720, 450674190, 658545360, 947426925, 1343646044, 1880535825, 2599922780, 3553856649, 4806611060

Offset: 1

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 = 1..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 (10,-45,120,-210,252,-210,120,-45,10,-1).

Crossrefs

Column 4 of A342985.

Programs

Mathematica

A006429[n_] := If[n == 1, 0, (n*(n + 2)*(n*(n*(n*(n*(n*(n*(13*n + 268) + 2254) + 4900) - 10703) - 62048) + 28596) + 137520))/60480];
Array[A006429, 50] (* Paolo Xausa, Aug 20 2025 *)

PARI

a(n) = if(n < 2, 0, n*(n+2)*(13*n^7+268*n^6+2254*n^5+4900*n^4-10703*n^3-62048*n^2+28596*n+137520) / 60480) \\ Andrew Howroyd, Apr 03 2021

Formula

a(n) = n*(n+2)*(13*n^7+268*n^6+2254*n^5+4900*n^4-10703*n^3-62048*n^2+28596*n+137520) / 60480 for n > 1. - Sean A. Irvine, Apr 10 2017

From Chai Wah Wu, Aug 01 2021: (Start)

a(n) = 10*a(n-1) - 45*a(n-2) + 120*a(n-3) - 210*a(n-4) + 252*a(n-5) - 210*a(n-6) + 120*a(n-7) - 45*a(n-8) + 10*a(n-9) - a(n-10) for n > 11.

G.f.: x^2*(-5*x^9 + 50*x^8 - 224*x^7 + 590*x^6 - 995*x^5 + 1100*x^4 - 735*x^3 + 198*x^2 + 95*x + 4)/(x - 1)^10. (End)

Extensions

Title improved by Sean A. Irvine, Apr 10 2017

Terms a(12) and beyond from Andrew Howroyd, Apr 03 2021