A342985 -id:A342985 - cp's OEIS frontend

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.

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.

A342987 Triangle read by rows: T(n,k) is the number of tree-rooted planar maps with n edges, k faces and no isthmuses, n >= 0, k = 1..n+1.

Original entry on oeis.org

1, 0, 1, 0, 2, 2, 0, 3, 15, 5, 0, 4, 60, 84, 14, 0, 5, 175, 650, 420, 42, 0, 6, 420, 3324, 5352, 1980, 132, 0, 7, 882, 13020, 42469, 37681, 9009, 429, 0, 8, 1680, 42240, 246540, 429120, 239752, 40040, 1430, 0, 9, 2970, 118998, 1142622, 3462354, 3711027, 1421226, 175032, 4862

Offset: 0

Andrew Howroyd, Apr 03 2021

The number of vertices is n + 2 - k.

For k >= 2, column k is a polynomial of degree 4*(k-2)+1.

			Triangle begins:
  1;
  0, 1;
  0, 2,    2;
  0, 3,   15,     5;
  0, 4,   60,    84,     14;
  0, 5,  175,   650,    420,     42;
  0, 6,  420,  3324,   5352,   1980,    132;
  0, 7,  882, 13020,  42469,  37681,   9009,   429;
  0, 8, 1680, 42240, 246540, 429120, 239752, 40040, 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 VIIIb.

Columns k=1..4 are A000007, A000027, A006470, A006471.
Diagonals are A000108, A002740, A006432, A006433.
Row sums are A342988.
Cf. A342981, A342982, A342984, A342985.

\\ here G(n,y) is A342984 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)}
G(n,y)={my(g=F(n,y)); subst(g, x, serreverse(x*g^2))}
H(n)={my(g=G(n,y)-x, v=Vec(sqrt(serreverse(x/g^2)/x))); [Vecrev(t) | t<-v]}
{ my(T=H(8)); for(n=1, #T, print(T[n])) }

G.f.: A(x,y) satisfies A(x,y) = G(x*A(x,y)^2,y) where G(x,y) + x is the g.f. of A342984.

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.

A342986 Number of tree-rooted loopless planar maps with n edges and no isthmuses.

Original entry on oeis.org

1, 0, 2, 6, 44, 280, 2100, 16310, 133652, 1132992, 9895672, 88520520, 808057712, 7504219008, 70730676392, 675328163542, 6521495669380, 63612394972608, 626076210568200, 6211621325369992, 62077602307372720, 624488579671582880, 6320044589443116720, 64313288809475362888

Offset: 0

Andrew Howroyd, Apr 03 2021

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..500

Row sums of A342985.

Cf. A004304, A342988.

\\ here J(n) gives A004304 as g.f.
J(n)={my(g=(1-sqrt(1-4*x+O(x^3*x^n)))/(2*x), h=serconvol(g, (g-1)/x));sqrt(x/serreverse(x*h^2))}
seq(n)={my(g=J(n)-2*x, p=O(1)); while(serprec(p, x)<=n, p = subst(g, x, x*p^2)); Vec(p)}

G.f.: A(x) satisfies A(x) = G(x*A(x)^2) where G(x) + 2*x is the g.f. of A004304.

A006428 Number of loopless tree-rooted planar maps with 3 vertices and n faces and no isthmuses.

Original entry on oeis.org

0, 3, 36, 135, 360, 798, 1568, 2826, 4770, 7645, 11748, 17433, 25116, 35280, 48480, 65348, 86598, 113031, 145540, 185115, 232848, 289938, 357696, 437550, 531050, 639873, 765828, 910861, 1077060, 1266660, 1482048, 1725768, 2000526, 2309195, 2654820, 3040623

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 (6,-15,20,-15,6,-1).

Column 3 of A342985.

A006428[n_] := If[n == 1, 0, n*(n + 1)*(n*(n*(n + 6) + 11) - 42)/24];
Array[A006428, 50] (* Paolo Xausa, Aug 20 2025, after Andrew Howroyd *)

a(n) = if(n<2, 0, n*(n+1)*(n^3+6*n^2+11*n-42) / 24) \\ Andrew Howroyd, Apr 03 2021

a(n) seems to be divisible by n+1. - Ralf Stephan, Sep 01 2003
Conjecture (for n > 1): a(n) = n*(n+1)*(n^3+6*n^2+11*n-42) / 24. - Sean A. Irvine, Apr 10 2017
The above conjectures are true. - Andrew Howroyd, Apr 03 2021
From Chai Wah Wu, Aug 08 2022: (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) for n > 7.
G.f.: x^2*(2*x^5 - 12*x^4 + 30*x^3 - 36*x^2 + 18*x + 3)/(x - 1)^6. (End)

Title improved by Sean A. Irvine, Apr 10 2017

Terms a(13) and beyond from Andrew Howroyd, Apr 03 2021

A006429 Number of loopless tree-rooted planar maps with 4 vertices and n faces.

Original entry on oeis.org

0, 4, 135, 1368, 7350, 28400, 89073, 241220, 585057, 1301420, 2699125, 5282172, 9842430, 17584416, 30289835, 50530680, 81940901, 129557940, 200246795, 303220720, 450674190, 658545360, 947426925, 1343646044, 1880535825, 2599922780, 3553856649, 4806611060

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 (10,-45,120,-210,252,-210,120,-45,10,-1).

Column 4 of A342985.

A006429[n_] := If[n == 1, 0, (n*(n + 2)*(n*(n*(n*(n*(n*(n*(13*n + 268) + 2254) + 4900) - 10703) - 62048) + 28596) + 137520))/60480];
Array[A006429, 50] (* Paolo Xausa, Aug 20 2025 *)

a(n) = if(n < 2, 0, n*(n+2)*(13*n^7+268*n^6+2254*n^5+4900*n^4-10703*n^3-62048*n^2+28596*n+137520) / 60480) \\ Andrew Howroyd, Apr 03 2021

a(n) = n*(n+2)*(13*n^7+268*n^6+2254*n^5+4900*n^4-10703*n^3-62048*n^2+28596*n+137520) / 60480 for n > 1. - Sean A. Irvine, Apr 10 2017
From Chai Wah Wu, Aug 01 2021: (Start)
a(n) = 10*a(n-1) - 45*a(n-2) + 120*a(n-3) - 210*a(n-4) + 252*a(n-5) - 210*a(n-6) + 120*a(n-7) - 45*a(n-8) + 10*a(n-9) - a(n-10) for n > 11.
G.f.: x^2*(-5*x^9 + 50*x^8 - 224*x^7 + 590*x^6 - 995*x^5 + 1100*x^4 - 735*x^3 + 198*x^2 + 95*x + 4)/(x - 1)^10. (End)

Title improved by Sean A. Irvine, Apr 10 2017

Terms a(12) and beyond from Andrew Howroyd, Apr 03 2021

A006430 Number of loopless tree-rooted planar maps with 5 vertices and n faces and no isthmuses.

Original entry on oeis.org

0, 5, 360, 7350, 73700, 474588, 2292790, 9046807, 30676440, 92393015, 252872984, 639382605, 1512137536, 3377126024, 7176513960, 14599539314, 28575632350, 54036739617, 99069119952, 176618150000, 306965183268, 521265871700, 866527603370, 1412513294049

Offset: 1

N. J. A. Sloane

nonn

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 A342985.

A006430[n_] := If[n == 1, 0, (n*(n + 2)*(n + 3)*(n*(n*(n*(n*(n*(n*(n*(n*(n*(23*n + 963) + 17544) + 147952) + 481675) - 1052153) - 7850914) - 2900162) + 60869272) + 37067400) - 179920800))/79833600];
Array[A006430, 50] (* Paolo Xausa, Aug 20 2025 *)

a(n)={if(n<2, 0, n*(n + 2)*(n + 3)*(23*n^10 + 963*n^9 + 17544*n^8 + 147952*n^7 + 481675*n^6 - 1052153*n^5 - 7850914*n^4 - 2900162*n^3 + 60869272*n^2 + 37067400*n - 179920800)/(2*11!))} \\ Andrew Howroyd, Apr 03 2021

a(n) = n*(n + 2)*(n + 3)*(23*n^10 + 963*n^9 + 17544*n^8 + 147952*n^7 + 481675*n^6 - 1052153*n^5 - 7850914*n^4 - 2900162*n^3 + 60869272*n^2 + 37067400*n - 179920800)/(2*11!) for n > 1.

Title improved by Sean A. Irvine, Apr 10 2017

Terms a(11) and beyond from Andrew Howroyd, Apr 03 2021

cp's OEIS Frontend

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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A342987 Triangle read by rows: T(n,k) is the number of tree-rooted planar maps with n edges, k faces and no isthmuses, n >= 0, k = 1..n+1.

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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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A342986 Number of tree-rooted loopless planar maps with n edges and no isthmuses.

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A006428 Number of loopless tree-rooted planar maps with 3 vertices and n faces and no isthmuses.

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A006429 Number of loopless tree-rooted planar maps with 4 vertices and n faces.

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A006430 Number of loopless tree-rooted planar maps with 5 vertices and n faces and no isthmuses.

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