A006412 Number of nonseparable tree-rooted planar maps with n + 3 edges and 4 vertices.

4, 75, 604, 3150, 12480, 40788, 115500, 292578, 677820, 1459315, 2954952, 5679700, 10438272, 18449760, 31511880, 52213596, 84206100, 132543411, 204105220, 308116050, 456776320, 666022500, 956435220, 1354315950, 1892954700, 2614113099, 3569749200, 4824012424

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 A342984.

Cf. A006411, A006413.

Programs

Mathematica

A006412[n_] := Binomial[n + 5, 6]*(n + 3)*(n*(13*n + 57) + 14)/84;
Array[A006412, 30] (* Paolo Xausa, Aug 20 2025 *)

PARI

a(n) = {binomial(n+5,6)*(n + 3)*(13*n^2 + 57*n + 14)/84} \\ Andrew Howroyd, Apr 05 2021

Formula

a(n) = 4 * binomial(n + 4, 5) + 51 * binomial(n + 4, 6) + 163 * binomial(n + 4, 7) + 194 * binomial(n + 4, 8) + 78 * binomial(n + 4, 9). - Sean A. Irvine, Apr 03 2017

a(n) = binomial(n+5,6)*(n + 3)*(13*n^2 + 57*n + 14)/84. - Andrew Howroyd, Apr 05 2021

G.f.: x*(4 + 35*x + 34*x^2 + 5*x^3)/(1 - x)^10. - Stefano Spezia, Aug 19 2025

Extensions

Terms a(11) and beyond from Andrew Howroyd, Apr 05 2021