A006470 Number of tree-rooted planar maps with 3 faces and n vertices and no isthmuses. (Formerly M2075)

2, 15, 60, 175, 420, 882, 1680, 2970, 4950, 7865, 12012, 17745, 25480, 35700, 48960, 65892, 87210, 113715, 146300, 185955, 233772, 290950, 358800, 438750, 532350, 641277, 767340, 912485, 1078800, 1268520, 1484032, 1727880, 2002770, 2311575, 2657340, 3043287, 3472820, 3949530, 4477200, 5059810, 5701542, 6406785, 7180140

Offset

1,1

Comments

a(n) is the number of ordered rooted trees with n+3 non-root nodes that have 3 leaves; see A108838. - Joerg Arndt, Aug 18 2014

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Vincenzo Librandi, 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, Vol. 18, No. 3 (1975), pp. 222-259.

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Formula

a(n) = (n+1)*binomial(n+3, 4).

a(n) = A027789(n)/2.

From Zerinvary Lajos, Dec 14 2005: (Start)

a(n) = binomial(n+2, 2)*binomial(n+4, 3)/2;

G.f.: x*(2+3*x)/(1-x)^6. (End)

From Wesley Ivan Hurt, May 02 2015: (Start)

a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6).

a(n) = n*(n+1)^2*(n+2)*(n+3)/24. (End)

Sum_{n>=1} 1/a(n) = 61/3 - 2*Pi^2. - Jaume Oliver Lafont, Jul 15 2017

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2 - 16*log(2) + 5/3. - Amiram Eldar, Jan 28 2022

Maple

A006470:=n->(n+1)*binomial(n+3, 4): seq(A006470(n), n=1..50); # Wesley Ivan Hurt, May 02 2015

Mathematica

Table[n (n + 1)^2 (n + 2) (n + 3) / 24, {n, 50}] (* Vincenzo Librandi, May 03 2015 *)
LinearRecurrence[{6, -15, 20, -15, 6, -1}, {2, 15, 60, 175, 420, 882}, 50] (* Harvey P. Dale, Jul 18 2024 *)

Prog

(Magma) [(n+1)*Binomial(n+3, 4): n in [1..30]]; // Vincenzo Librandi, Jun 09 2013

Crossrefs

Column 3 of A342987.

Cf. A027789, A108838.

Sequence in context: A295828 A126019 A071237 * A084169 A337905 A296661

Adjacent sequences: A006467 A006468 A006469 * A006471 A006472 A006473

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

Name clarified by Andrew Howroyd, Apr 03 2021

Status

approved