A006411 Number of nonseparable tree-rooted planar maps with n + 2 edges and 3 vertices. (Formerly M3073)

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

1,1

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. See Table IVa.

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

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

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 *)

Prog

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

Crossrefs

Column 3 of A342984.

Cf. A006412, A006413, A027777.

Sequence in context: A289643 A196741 A196899 * A243208 A373499 A129549

Adjacent sequences: A006408 A006409 A006410 * A006412 A006413 A006414

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

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

Status

approved