A006412 Number of nonseparable tree-rooted planar maps with n + 3 edges and 4 vertices. (Formerly M3697)

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

1,1

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

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

Mathematica

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

Prog

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

Crossrefs

Column 4 of A342984.

Cf. A006411, A006413.

Sequence in context: A280889 A257367 A072373 * A206456 A386860 A137220

Adjacent sequences: A006409 A006410 A006411 * A006413 A006414 A006415

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

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

Status

approved