A006470 Number of tree-rooted planar maps with 3 faces and n vertices and no isthmuses.
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
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).
Programs
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Magma
[(n+1)*Binomial(n+3, 4): n in [1..30]]; // Vincenzo Librandi, Jun 09 2013
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Maple
A006470:=n->(n+1)*binomial(n+3,4): seq(A006470(n), n=1..50); # Wesley Ivan Hurt, May 02 2015
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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 *)
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
Extensions
Name clarified by Andrew Howroyd, Apr 03 2021
Comments