A006413 Number of nonseparable tree-rooted planar maps with n + 4 edges and 5 vertices.

5, 210, 3150, 27556, 170793, 829920, 3359356, 11786190, 36845718, 104719524, 274707420, 672982128, 1554007910, 3407724936, 7139933088, 14366348780, 27878652291, 52364814150, 95497666810, 169546939380, 293722986375, 497527759560, 825473130300, 1343631834090

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 (14,-91,364,-1001,2002,-3003,3432,-3003,2002,-1001,364,-91,14,-1).

Crossrefs

Column 5 of A342984.

Cf. A006411, A006412.

Programs

Mathematica

A006413[n_] := Binomial[n + 7, 8]*(n + 4)*(n*(n*(n*(23*n + 279) + 941) + 599) + 138)/1980;
Array[A006413, 25] (* Paolo Xausa, Aug 20 2025 *)

PARI

a(n) = {binomial(n+7, 8)*(n + 4)*(23*n^4 + 279*n^3 + 941*n^2 + 599*n + 138)/1980} \\ Andrew Howroyd, Apr 05 2021

Formula

a(n) = 5 * binomial(n + 6, 7) + 170 * binomial(n + 6, 8) + 1440 * binomial(n + 6, 9) + 4906 * binomial(n + 6, 10) + 7927 * binomial(n + 6, 11) + 6090 * binomial(n + 6, 12) + 1794 * binomial(n + 6, 13). - Sean A. Irvine, Apr 03 2017

a(n) = binomial(n+7,8)*(n + 4)*(23*n^4 + 279*n^3 + 941*n^2 + 599*n + 138)/1980. - Andrew Howroyd, Apr 05 2021

G.f.: x*(5 + 140*x + 665*x^2 + 746*x^3 + 224*x^4 + 14*x^5)/(1 - x)^14. - Stefano Spezia, Aug 19 2025

Extensions

Terms a(10) and beyond from Andrew Howroyd, Apr 05 2021