A006471 Number of tree-rooted planar maps with 4 faces and n vertices and no isthmuses.

5, 84, 650, 3324, 13020, 42240, 118998, 300300, 693693, 1490060, 3011580, 5779592, 10608000, 18728832, 31957620, 52907400, 85261341, 134115300, 206402966, 311417700, 461446700, 672534720, 965396250, 1366496820, 1909325925, 2635885980, 3598423704, 4861432400, 6503955744, 8622225920

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
• 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 A342987.

Programs

Mathematica

A006471[n_] := n*(n + 1)*(n + 2)^2*(n + 3)*(n + 4)*(n + 5)*(n*(13*n + 73) + 54)/60480;
Array[A006471, 50] (* Paolo Xausa, Aug 20 2025 *)

PARI

a(n) = (n*(2+n)^2*(3240 +10158*n +11777*n^2 +6400*n^3 +1770*n^4 +242*n^5 +13*n^6))/60480 \\ Andrew Howroyd, Apr 03 2021

Formula

From Colin Barker, Apr 09 2013: (Start)

a(n) = (n*(2+n)^2*(3240 + 10158*n + 11777*n^2 + 6400*n^3 + 1770*n^4 + 242*n^5 + 13*n^6))/60480.

G.f.: x*(4*x^3 + 35*x^2 + 34*x + 5) / (x-1)^10. (End)

Extensions

Name clarified and terms a(12) and beyond from Andrew Howroyd, Apr 03 2021