A006411 Number of nonseparable tree-rooted planar maps with n + 2 edges and 3 vertices.

3, 20, 75, 210, 490, 1008, 1890, 3300, 5445, 8580, 13013, 19110, 27300, 38080, 52020, 69768, 92055, 119700, 153615, 194810, 244398, 303600, 373750, 456300, 552825, 665028, 794745, 943950, 1114760, 1309440, 1530408, 1780240, 2061675, 2377620, 2731155, 3125538

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

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

Crossrefs

Column 3 of A342984.

Cf. A006412, A006413, A027777.

Programs

Magma

[n*(n+1)*(n+2)^2*(n+3)/24: n in [1..50]]; // Vincenzo Librandi, May 19 2011

Maple

A006411:=n->n*(n+1)*(n+2)^2*(n+3)/24: seq(A006411(n), n=1..50); # Wesley Ivan Hurt, Jul 15 2017

Mathematica

CoefficientList[Series[x (3+2x)/(1-x)^6,{x,0,40}],x] (* or *) LinearRecurrence[ {6,-15,20,-15,6,-1},{0,3,20,75,210,490},40] (* Harvey P. Dale, Dec 24 2013 *)

Formula

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

a(n) = n*(n+1)*(n+2)^2*(n+3)/24. - Bruno Berselli, May 17 2011

a(n) = A027777(n)/2. - Zerinvary Lajos, Mar 23 2007

a(n) = binomial(n+2,n)*binomial(n+2,n-1) - binomial(n+2,n+1)*binomial(n+2,n-2). - J. M. Bergot, Apr 07 2013

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). - Harvey P. Dale, Dec 24 2013

Sum_{n>=1} 1/a(n) = 2*Pi^2 - 58/3. - Jaume Oliver Lafont, Jul 15 2017

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2 + 16*log(2) - 62/3. - Amiram Eldar, Jan 28 2022

Extensions

G.f. adapted to the offset by Bruno Berselli, May 17 2011