A006471 Number of tree-rooted planar maps with 4 faces and n vertices and no isthmuses. (Formerly M4021)

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

1,1

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

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).

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)

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 *)

Prog

(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

Crossrefs

Column 4 of A342987.

Sequence in context: A271219 A359110 A187587 * A383451 A258391 A061628

Adjacent sequences: A006468 A006469 A006470 * A006472 A006473 A006474

Keyword

nonn

Author

N. J. A. Sloane

Extensions

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

Status

approved