A269920 -id:A269920 - cp's OEIS frontend

A342981 Triangle read by rows: T(n,k) is the number of rooted planar maps with n edges, k faces and no isthmuses, n >= 0, k = 1..n+1.

1, 0, 1, 0, 1, 2, 0, 1, 7, 5, 0, 1, 16, 37, 14, 0, 1, 30, 150, 176, 42, 0, 1, 50, 449, 1104, 794, 132, 0, 1, 77, 1113, 4795, 7077, 3473, 429, 0, 1, 112, 2422, 16456, 41850, 41504, 14893, 1430, 0, 1, 156, 4788, 47832, 189183, 319320, 228810, 63004, 4862

Offset: 0

Andrew Howroyd, Apr 02 2021

The number of vertices is n + 2 - k.
For k >= 2, column k is a polynomial of degree 3*(k-2). This is because adding a face can increase the number of vertices whose degree is greater than two by at most two.
By duality, also the number of loopless rooted planar maps with n edges and k vertices.

			Triangle begins:
  1;
  0, 1;
  0, 1,   2;
  0, 1,   7,    5;
  0, 1,  16,   37,    14;
  0, 1,  30,  150,   176,    42;
  0, 1,  50,  449,  1104,   794,   132;
  0, 1,  77, 1113,  4795,  7077,  3473,   429;
  0, 1, 112, 2422, 16456, 41850, 41504, 14893, 1430;
  ...

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 VIb.

Columns k=3..4 are A005581, A006468.
Diagonals are A000108, A006419, A006420, A006421.
Row sums are A000260.
Cf. A002293, A027836, A082680, A269920, A342980, A343092.

G[m_, y_] := Sum[x^n*Sum[(n + k - 1)!*(2*n - k)!*y^k/(k!*(n + 1 - k)!*(2*k - 1)!*(2*n - 2*k + 1)!), {k, 1, n}], {n, 1, m}] + O[x]^m;
H[n_] := With[{g = 1 + x*y + x*G[n - 1, y]}, Sqrt[InverseSeries[x/g^2 + O[x]^(n + 1), x]/x]];
CoefficientList[#, y]& /@ CoefficientList[H[10], x] // Flatten (* Jean-François Alcover, Apr 15 2021, after Andrew Howroyd *)

\\ here G(n, y) gives A082680 as g.f.
G(n,y)={sum(n=1, n, x^n*sum(k=1, n, (n+k-1)!*(2*n-k)!*y^k/(k!*(n+1-k)!*(2*k-1)!*(2*n-2*k+1)!))) + O(x*x^n)}
H(n)={my(g=1+x*y+x*G(n-1, y), v=Vec(sqrt(serreverse(x/g^2)/x))); vector(#v, n, Vecrev(v[n], n))}
{ my(T=H(8)); for(n=1, #T, print(T[n])) }

G.f. A(x,y) satisfies A(x) = G(x*A(x,y)^2, y) where G(x,y) = 1 + x*y + x*B(x,y) and B(x,y) is the g.f. of A082680.
A027836(n+1) = Sum_{k=1..n+1} k*T(n,k).
A002293(n) = Sum_{k=1..n+1} k*T(n,n+2-k).

A277741 Array read by antidiagonals: A(n,k) is the number of unsensed planar maps with n vertices and k faces, n >= 1, k >= 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 5, 5, 2, 3, 13, 20, 13, 3, 6, 35, 83, 83, 35, 6, 12, 104, 340, 504, 340, 104, 12, 27, 315, 1401, 2843, 2843, 1401, 315, 27, 65, 1021, 5809, 15578, 21420, 15578, 5809, 1021, 65, 175, 3407, 24299, 82546, 149007, 149007, 82546, 24299, 3407, 175

Offset: 1

N. J. A. Sloane, Nov 07 2016

A(n,k) is also the number of multiquadrangulations of the sphere with n stable equilibria and k unstable equilibria.
From Andrew Howroyd, Jan 13 2025: (Start)
The planar maps considered are connected and may contain loops and parallel edges.
The number of edges is n + k - 2. (End)

			The array begins:
   1,    1,    1,     2,     3,     6,   12,   27, 65, ...
   1,    2,    5,    13,    35,   104,  315, 1021, ...
   1,    5,   20,    83,   340,  1401, 5809, ...
   2,   13,   83,   504,  2843, 15578, ...
   3,   35,  340,  2843, 21420, ...
   6,  104, 1401, 15578, ...
  12,  315, 5809, ...
  27, 1021, ...
  65, ...
  ...
As a triangle, rows give the number of edges (first row is 0 edges):
   1;
   1,    1;
   1,    2,    1;
   2,    5,    5,     2;
   3,   13,   20,    13,     3;
   6,   35,   83,    83,    35,    6;
  12,  104,  340,   504,   340,   104,   12;
  27,  315, 1401,  2843,  2843,  1401,  315,   27;
  65, 1021, 5809, 15578, 21420, 15578, 5809, 1021, 65;
  ...

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, chapter 5.

Richard Kapolnai, Gabor Domokos, and Timea Szabo, Generating spherical multiquadrangulations by restricted vertex splittings and the reducibility of equilibrium classes, Periodica Polytechnica Electrical Engineering, 56(1):11-10, 2012. Also arXiv:1206.1698 [cs.DM], 2012. See Table 1.
Timothy R. Walsh, Number of sensed planar maps with n edges and m vertices, pp. 11-20.
Nicholas C. Wormald, Counting unrooted planar maps, Discrete Math. 36 (1981), no. 2, 205-225.

Antidiagonal sums are A006385.
Rows 1..2 (equally, columns 1..2) are A006082, A380239.
Cf. A269920 (rooted), A379430 (sensed), A379431 (achiral), A379432 (2-connected), A384963 (simple).

A(n,k) = A(k,n).

A(n,k) = (A379430(n,k) + A379431(n,k))/2. - Andrew Howroyd, Jan 14 2025

Missing terms inserted and definition edited by Andrew Howroyd, Jan 13 2025

A342980 Triangle read by rows: T(n,k) is the number of rooted loopless planar maps with n edges, k faces and no isthmuses, n >= 0, k = 1..n+1.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 8, 1, 0, 0, 1, 20, 20, 1, 0, 0, 1, 38, 131, 38, 1, 0, 0, 1, 63, 469, 469, 63, 1, 0, 0, 1, 96, 1262, 3008, 1262, 96, 1, 0, 0, 1, 138, 2862, 12843, 12843, 2862, 138, 1, 0, 0, 1, 190, 5780, 42602, 83088, 42602, 5780, 190, 1, 0

Offset: 0

Andrew Howroyd, Apr 01 2021

The number of vertices is n + 2 - k.

For k >= 2, columns k without the initial zero term is a polynomial of degree 3*(k-2). This is because adding a face can increase the number of vertices whose degree is greater than two by at most two.

			Triangle begins:
  1;
  0, 0;
  0, 1,   0;
  0, 1,   1,    0;
  0, 1,   8,    1,     0;
  0, 1,  20,   20,     1,     0;
  0, 1,  38,  131,    38,     1,    0;
  0, 1,  63,  469,   469,    63,    1,   0;
  0, 1,  96, 1262,  3008,  1262,   96,   1, 0;
  0, 1, 138, 2862, 12843, 12843, 2862, 138, 1, 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 VIa.

Columns (and diagonals) are A006416, A006417, A006418.
Row sums are A099553(n+1).
Cf. A082680, A269920, A342981, A342985, A343090.

G[m_, y_] := Sum[x^n*Sum[(n + k - 1)!*(2*n - k)!*y^k/(k!*(n + 1 - k)!*(2*k - 1)!*(2*n - 2*k + 1)!), {k, 1, n}], {n, 1, m}] + O[x]^m;
H[n_] := With[{g = 1 + x*G[n - 1, y]}, Sqrt[InverseSeries[x/g^2 + O[x]^(n + 1), x]/x]];
Join[{{1}, {0, 0}}, Append[CoefficientList[#, y], 0]& /@ CoefficientList[ H[11], x][[3;;]]] // Flatten (* Jean-François Alcover, Apr 15 2021, after Andrew Howroyd *)

\\ here G(n,y) gives A082680 as g.f.
G(n,y)={sum(n=1, n, x^n*sum(k=1, n, (n+k-1)!*(2*n-k)!*y^k/(k!*(n+1-k)!*(2*k-1)!*(2*n-2*k+1)!))) + O(x*x^n)}
H(n)={my(g=1+x*G(n-1, y), v=Vec(sqrt(serreverse(x/g^2)/x))); 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 A(x,y) = G(x*A(x,y)^2,y) where G(x,y) = 1 + x*B(x,y) and B(x,y) is the g.f. of A082680.

A379430 Array read by antidiagonals: A(n,k) is the number of sensed planar maps with n vertices and k faces, n >= 1, k >= 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 5, 5, 2, 3, 14, 23, 14, 3, 6, 42, 108, 108, 42, 6, 14, 140, 501, 761, 501, 140, 14, 34, 473, 2264, 4744, 4744, 2264, 473, 34, 95, 1670, 10087, 27768, 38495, 27768, 10087, 1670, 95, 280, 5969, 44310, 153668, 279698, 279698, 153668, 44310, 5969, 280

Offset: 1

Andrew Howroyd, Jan 13 2025

The planar maps considered are connected and may contain loops and parallel edges.

The number of edges is n + k - 2.

			Array begins:
=========================================================
n\k |  1    2     3      4      5      6      7     8 ...
----+----------------------------------------------------
  1 |  1    1     1      2      3      6     14    34 ...
  2 |  1    2     5     14     42    140    473  1670 ...
  3 |  1    5    23    108    501   2264  10087 44310 ...
  4 |  2   14   108    761   4744  27768 153668 ...
  5 |  3   42   501   4744  38495 279698 ...
  6 |  6  140  2264  27768 279698 ...
  7 | 14  473 10087 153668 ...
  8 | 34 1670 44310 ...
   ...
As a triangle, rows give the number of edges (first row is 0 edges):
   1;
   1,    1;
   1,    2,     1;
   2,    5,     5,     2;
   3,   14,    23,    14,     3;
   6,   42,   108,   108,    42,     6;
  14,  140,   501,   761,   501,   140,    14;
  34,  473,  2264,  4744,  4744,  2264,   473,   34;
  95, 1670, 10087, 27768, 38495, 27768, 10087, 1670, 95;
  ...

Timothy R. Walsh, Number of sensed planar maps with n edges and m vertices, pp. 1-10.

Antidiagonal sums are A006384.
Columns 1..2 are A002995, A380237.
Cf. A269920 (rooted), A277741 (unsensed), A379431 (achiral), A342061 (2-connected), A384964 (simple).

A(n,k) = A(k,n).

A342982 Triangle read by rows: T(n,k) is the number of tree-rooted planar maps with n edges and k+1 faces, n >= 0, k = 0..n.

Original entry on oeis.org

1, 1, 1, 2, 6, 2, 5, 30, 30, 5, 14, 140, 280, 140, 14, 42, 630, 2100, 2100, 630, 42, 132, 2772, 13860, 23100, 13860, 2772, 132, 429, 12012, 84084, 210210, 210210, 84084, 12012, 429, 1430, 51480, 480480, 1681680, 2522520, 1681680, 480480, 51480, 1430

Offset: 0

Andrew Howroyd, Apr 03 2021

The number of vertices is n + 1 - k.

A tree-rooted planar map is a planar map with a distinguished spanning tree.

			Triangle begins:
    1;
    1,     1;
    2,     6,     2;
    5,    30,    30,      5;
   14,   140,   280,    140,     14;
   42,   630,  2100,   2100,    630,    42;
  132,  2772, 13860,  23100,  13860,  2772,   132;
  429, 12012, 84084, 210210, 210210, 84084, 12012, 429;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
R. C. Mullin, On the Enumeration of Tree-Rooted Maps, Canadian Journal of Mathematics, Volume 19, 1967, pp. 174-183.

Columns k=0..2 are A000108, A002457, 2*A002803.
Row sums are A005568.
Central coefficients are A342983.
Cf. A001263, A215288, A269920, A342984.

Table[(2 n)!/(k!*(k + 1)!*(n - k)!*(n - k + 1)!), {n, 0, 8}, {k, 0, n}] // Flatten (* Michael De Vlieger, Apr 06 2021 *)

T(n,k) = {(2*n)!/(k!*(k+1)!*(n-k)!*(n-k+1)!)}
{ for(n=0, 10, print(vector(n+1, k, T(n,k-1)))) }

T(n,k) = (2*n)!/(k!*(k+1)!*(n-k)!*(n-k+1)!).
T(n,n-k) = T(n,k).
T(n, floor(n/2)) = A215288(n).
T(n,k) = A000108(n) * A001263(n+1,k+1). - Werner Schulte, Apr 04 2021

A000184 Number of genus 0 rooted maps with 3 faces with n vertices.

Original entry on oeis.org

2, 22, 164, 1030, 5868, 31388, 160648, 795846, 3845020, 18211380, 84876152, 390331292, 1775032504, 7995075960, 35715205136, 158401506118, 698102372988, 3059470021316, 13341467466520, 57918065919924, 250419305769512, 1078769490401032, 4631680461623664, 19825379450255900, 84622558822506328, 360270317908904328, 1530148541536781488, 6484511936352543096, 27423786092731382000, 115756362341775227888

Offset: 2

N. J. A. Sloane

nonn

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
T. R. S. Walsh, Combinatorial Enumeration of Non-Planar Maps. Ph.D. Dissertation, Univ. of Toronto, 1971.

Alois P. Heinz, Table of n, a(n) for n = 2..500
Richard P. Stanley, CATALAN ADDENDUM, version of Jul 19, 2008, p. 24. [From _Jonathan Vos Post_, Aug 16 2008]
M. S. Tokmachev, Correlations Between Elements and Sequences in a Numerical Prism, Bulletin of the South Ural State University, Ser. Mathematics. Mechanics. Physics, 2019, Vol. 11, No. 1, 24-33.
W. T. Tutte, On the enumeration of planar maps, Bull. Amer. Math. Soc. 74 1968 64-74.
T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus, J. Comb. Thy B13 (1972), 122-141 and 192-218.
Notes

Column 3 of A269920.
Column 0 of A270407.
Cf. A000108, A029887.

[n*((n+1)*(n+2)*Catalan(n+1) - 3*4^n)/12: n in [2..30]]; // G. C. Greubel, Jul 18 2024

a[n_] := 1/12*(2^(n+1)*(2*n+1)!!/(n-1)!-3*4^n*n); Table[a[n], {n, 2, 31}] (* Jean-François Alcover, Mar 12 2014 *)

[n*(2*(2*n+1)*binomial(2*n,n) - 3*4^n)//12 for n in range(2,30)] # G. C. Greubel, Jul 18 2024

a(n) = 2 * A029887(n-2). - Ralf Stephan, Aug 17 2004
a(n) = 4^n*Gamma(n+3/2)/(3*sqrt(Pi)*Gamma(n)) - n*4^(n-1). - Mark van Hoeij, Jul 06 2010
From G. C. Greubel, Jul 18 2024: (Start)
a(n) = (n/12)*( (n+1)*(n+2)*Catalan(n+1) - 3*4^n ).
G.f.: x*(1 - sqrt(1 - 4*x))/(1-4*x)^(5/2).
E.g.f.: (x/3)*exp(2*x)*( - 3*exp(2*x) + 3*(1+2*x)*BesselI(0, 2*x) + (3+8*x)*BesselI(1, 2*x) + 2*x*BesselI(2, 2*x) ). (End)

More terms from Sean A. Irvine, Nov 14 2010

A379431 Array read by antidiagonals: A(n,k) is the number of achiral planar maps with n vertices and k faces, n >= 1, k >= 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 5, 5, 2, 3, 12, 17, 12, 3, 6, 28, 58, 58, 28, 6, 10, 68, 179, 247, 179, 68, 10, 20, 157, 538, 942, 942, 538, 157, 20, 35, 372, 1531, 3388, 4345, 3388, 1531, 372, 35, 70, 845, 4288, 11424, 18316, 18316, 11424, 4288, 845, 70

Offset: 1

Andrew Howroyd, Jan 14 2025

The planar maps considered are connected and may contain loops and parallel edges.

The number of edges is n + k - 2.

			==================================================
n\k |  1   2    3     4     5     6     7    8 ...
----+---------------------------------------------
  1 |  1   1    1     2     3     6    10   20 ...
  2 |  1   2    5    12    28    68   157  372 ...
  3 |  1   5   17    58   179   538  1531 4288 ...
  4 |  2  12   58   247   942  3388 11424 ...
  5 |  3  28  179   942  4345 18316 ...
  6 |  6  68  538  3388 18316 ...
  7 | 10 157 1531 11424 ...
  8 | 20 372 4288 ...
  ...
As a triangle, rows give the number of edges (first row is 0 edges):
   1;
   1,   1;
   1,   2,    1;
   2,   5,    5,   2;
   3,  12,   17,   12,    3;
   6,  28,   58,   58,   28,    6;
  10,  68,  179,  247,  179,   68,   10;
  20, 157,  538,  942,  942,  538,  157,  20;
  35, 372, 1531, 3388, 4345, 3388, 1531, 372, 35;
  ...

Antidiagonal sums are A006443.
Column 1 is A210736(n-1).
Cf. A269920 (rooted), A277741 (unsensed), A379430 (sensed).

A(n,k) = A(k,n).

A000365 Number of genus 0 rooted planar maps with 4 faces and n vertices.

Original entry on oeis.org

5, 93, 1030, 8885, 65954, 442610, 2762412, 16322085, 92400330, 505403910, 2687477780, 13957496098, 71053094420, 355548314180, 1752827693528, 8529176056965, 41026491589722, 195327793313790, 921451498774660, 4311086414580022, 20019238138410940

Offset: 3

N. J. A. Sloane

nonn

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
T. R. S. Walsh, Combinatorial Enumeration of Non-Planar Maps. Ph.D. Dissertation, Univ. of Toronto, 1971.

T. D. Noe, Table of n, a(n) for n = 3..200
W. T. Tutte, On the enumeration of planar maps, Bull. Amer. Math. Soc. 74 1968 64-74.
T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus, J. Comb. Thy B13 (1972), 122-141 and 192-218.
Notes

Column 4 of A269920.

Column 0 of A270408.

nn = 20; CoefficientList[Series[x^2 (1 - Sqrt[1 - 4 x]) (7 + 4 x - 2 Sqrt[1 - 4 x])/(2 (4 x - 1)^4), {x, 0, nn}], x] (* T. D. Noe, Jun 19 2012 *)

seq(n)={my(g=sqrt(1-4*x + O(x*x^n))); Vec((1-g)*(7+4*x-2*g)/(2*(1-4*x)^4))} \\ Andrew Howroyd, Mar 27 2021

G.f.: x^2*(1-sqrt(1-4*x))*(7+4*x-2*sqrt(1-4*x))/(2*(4*x-1)^4). - corrected for right offset by Vaclav Kotesovec, Aug 13 2013

a(n) ~ n^3*4^n/24 * (1-4/(sqrt(Pi*n))). - Vaclav Kotesovec, Aug 13 2013

More terms from Sean A. Irvine, Nov 14 2010

A380241 Array read by antidiagonals: T(n,k) is the number of rooted (2k)-regular planar maps with n vertices, n >= 0, k >= 0.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 9, 1, 1, 1, 14, 100, 54, 1, 1, 1, 42, 1225, 3000, 378, 1, 1, 1, 132, 15876, 171500, 110000, 2916, 1, 1, 1, 429, 213444, 10001880, 30012500, 4550000, 24057, 1, 1, 1, 1430, 2944656, 591666768, 7981500240, 5987493750, 204000000, 208494, 1, 1

Offset: 0

Andrew Howroyd, Jan 22 2025

The zeroth column is included by convention only for consistency with the first row sequences.

The case for regular planar maps of odd valency is more complicated and without simple closed form formulas, so not presented in this sequence. See the references for additional information.

			Array begins:
====================================================================
n\k | 0  1      2          3               4                   5 ...
----+---------------------------------------------------------------
  0 | 1  1      1          1               1                   1 ...
  1 | 1  1      2          5              14                  42 ...
  2 | 1  1      9        100            1225               15876 ...
  3 | 1  1     54       3000          171500            10001880 ...
  4 | 1  1    378     110000        30012500          7981500240 ...
  5 | 1  1   2916    4550000      5987493750       7304332956480 ...
  6 | 1  1  24057  204000000   1302227368750    7310748066293952 ...
  7 | 1  1 208494 9690000000 301107909375000 7794097754539041792 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals)
E. A. Bender and E. R. Canfield, The number of degree restricted rooted maps on the sphere, SIAM J. Discrete Math. 7 (1994) 9-15.
Zhicheng Gao and Mizan Rahman, Enumeration of k-poles, Annals of Combinatorics 1 (1997), pp. 55-66.
W. T. Tutte, A Census of Slicings, Canad. J. Math. 14 (1962), 708-722.
W. T. Tutte, A Census of Planar Maps, Canad. J. Math. 15 (1963), 249-271.

Columns 0..3 are A000012 twice, A000168, A380242.
Rows 0..3 are A000012, A000108, A060150, A380243.
Cf. A269920.

T(n,k)=if(k==0, 1, 2*binomial(2*k-1,k)^n*(n*k)!/(n!*(n*k - n + 2)!))

T(n,k) = 2*binomial(2*k-1, k)^n*(n*k)!/(n!*(n*k - n + 2)!) for k > 0.

A000473 Number of genus 0 rooted maps with 5 faces and n vertices.

Original entry on oeis.org

14, 386, 5868, 65954, 614404, 5030004, 37460376, 259477218, 1697186964, 10596579708, 63663115880, 370293754740, 2095108370600, 11574690111400, 62629794691632, 332742342741090, 1739371969822260, 8961709528660140, 45576855706440520, 229087231033907708

Offset: 4

N. J. A. Sloane

nonn

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
T. R. S. Walsh, Combinatorial Enumeration of Non-Planar Maps. Ph.D. Dissertation, Univ. of Toronto, 1971.

Andrew Howroyd, Table of n, a(n) for n = 4..500
W. T. Tutte, On the enumeration of planar maps, Bull. Amer. Math. Soc. 74 1968 64-74.
T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus, J. Comb. Thy B13 (1972), 122-141 and 192-218.
Notes

Column 5 of A269920.

Column 0 of A270409.

CoefficientList[(1/x)(1-Sqrt[1-4x])(17+16x-(10+4x)Sqrt[1-4x])/(1-4x)^(11/2) + O[x]^36, x] (* Jean-François Alcover, Feb 08 2016 *)

seq(n)={my(g=sqrt(1-4*x + O(x*x^n))); Vec((1-g)*(17+16*x-(10+4*x)*g)/((1-4*x)^5*g))} \\ Andrew Howroyd, Mar 28 2021

G.f.: x^3*(1-sqrt(1-4*x))*(17+16*x-(10+4*x)*sqrt(1-4*x))/(1-4*x)^(11/2). - Sean A. Irvine, Nov 14 2010

More terms from Sean A. Irvine, Nov 14 2010

cp's OEIS Frontend

A342981 Triangle read by rows: T(n,k) is the number of rooted planar maps with n edges, k faces and no isthmuses, n >= 0, k = 1..n+1.

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A277741 Array read by antidiagonals: A(n,k) is the number of unsensed planar maps with n vertices and k faces, n >= 1, k >= 1.

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A342980 Triangle read by rows: T(n,k) is the number of rooted loopless planar maps with n edges, k faces and no isthmuses, n >= 0, k = 1..n+1.

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A379430 Array read by antidiagonals: A(n,k) is the number of sensed planar maps with n vertices and k faces, n >= 1, k >= 1.

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A342982 Triangle read by rows: T(n,k) is the number of tree-rooted planar maps with n edges and k+1 faces, n >= 0, k = 0..n.

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A000184 Number of genus 0 rooted maps with 3 faces with n vertices.

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A379431 Array read by antidiagonals: A(n,k) is the number of achiral planar maps with n vertices and k faces, n >= 1, k >= 1.

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A000365 Number of genus 0 rooted planar maps with 4 faces and n vertices.

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