A006412 -id:A006412 - cp's OEIS frontend

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

N. J. A. Sloane

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

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).

Column 3 of A342984.

Cf. A006412, A006413, A027777.

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

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

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 *)

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

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

A342984 Triangle read by rows: T(n,k) is the number of nonseparable tree-rooted planar maps with n edges and k faces, n >= 0, k = 1..n+1.

Original entry on oeis.org

1, 1, 1, 0, 2, 0, 0, 3, 3, 0, 0, 4, 20, 4, 0, 0, 5, 75, 75, 5, 0, 0, 6, 210, 604, 210, 6, 0, 0, 7, 490, 3150, 3150, 490, 7, 0, 0, 8, 1008, 12480, 27556, 12480, 1008, 8, 0, 0, 9, 1890, 40788, 170793, 170793, 40788, 1890, 9, 0, 0, 10, 3300, 115500, 829920, 1565844, 829920, 115500, 3300, 10, 0

Offset: 0

Andrew Howroyd, Apr 03 2021

The number of vertices is n + 2 - k.
A tree-rooted planar map is a planar map with a distinguished spanning tree.
For k >= 2, column k is a polynomial of degree 4*(k-2)+1.

			Triangle begins:
  1;
  1, 1;
  0, 2,    0;
  0, 3,    3,     0;
  0, 4,   20,     4,     0;
  0, 5,   75,    75,     5,     0;
  0, 6,  210,   604,   210,     6,    0;
  0, 7,  490,  3150,  3150,   490,    7, 0;
  0, 8, 1008, 12480, 27556, 12480, 1008, 8, 0;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
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, Table IVa.

Columns (and diagonals) 3..5 are A006411, A006412, A006413.
Row sums are A004304.
Cf. A342982, A342985, A342987.

\\ here F(n,y) gives A342982 as g.f.
F(n,y)={sum(n=0, n, x^n*sum(i=0, n, my(j=n-i); y^i*(2*i+2*j)!/(i!*(i+1)!*j!*(j+1)!))) + O(x*x^n)}
H(n)={my(g=F(n,y), v=Vec(subst(g, x, serreverse(x*g^2)))); vector(#v, n, Vecrev(v[n], n))}
{ my(T=H(8)); for(n=1, #T, print(T[n])) }

T(n,n+2-k) = T(n,k).

G.f. A(x,y) satisfies F(x,y) = A(x*F(x,y)^2,y) where F(x,y) is the g.f. of A342982.

A006413 Number of nonseparable tree-rooted planar maps with n + 4 edges and 5 vertices.

Original entry on oeis.org

5, 210, 3150, 27556, 170793, 829920, 3359356, 11786190, 36845718, 104719524, 274707420, 672982128, 1554007910, 3407724936, 7139933088, 14366348780, 27878652291, 52364814150, 95497666810, 169546939380, 293722986375, 497527759560, 825473130300, 1343631834090

Offset: 1

N. J. A. Sloane

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

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 (14,-91,364,-1001,2002,-3003,3432,-3003,2002,-1001,364,-91,14,-1).

Column 5 of A342984.

Cf. A006411, A006412.

A006413[n_] := Binomial[n + 7, 8]*(n + 4)*(n*(n*(n*(23*n + 279) + 941) + 599) + 138)/1980;
Array[A006413, 25] (* Paolo Xausa, Aug 20 2025 *)

a(n) = {binomial(n+7, 8)*(n + 4)*(23*n^4 + 279*n^3 + 941*n^2 + 599*n + 138)/1980} \\ Andrew Howroyd, Apr 05 2021

a(n) = 5 * binomial(n + 6, 7) + 170 * binomial(n + 6, 8) + 1440 * binomial(n + 6, 9) + 4906 * binomial(n + 6, 10) + 7927 * binomial(n + 6, 11) + 6090 * binomial(n + 6, 12) + 1794 * binomial(n + 6, 13). - Sean A. Irvine, Apr 03 2017
a(n) = binomial(n+7,8)*(n + 4)*(23*n^4 + 279*n^3 + 941*n^2 + 599*n + 138)/1980. - Andrew Howroyd, Apr 05 2021
G.f.: x*(5 + 140*x + 665*x^2 + 746*x^3 + 224*x^4 + 14*x^5)/(1 - x)^14. - Stefano Spezia, Aug 19 2025

Terms a(10) and beyond from Andrew Howroyd, Apr 05 2021

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

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A342984 Triangle read by rows: T(n,k) is the number of nonseparable tree-rooted planar maps with n edges and k faces, n >= 0, k = 1..n+1.

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A006413 Number of nonseparable tree-rooted planar maps with n + 4 edges and 5 vertices.

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