A082680 -id:A082680 - 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).

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.

A278880 Triangle where g.f. S = S(x,m) satisfies: S = x/(G(-S^2)G(-mS^2)) such that G(x) = 1 + xG(x)^2 is the g.f. of the Catalan numbers (A000108), as read by rows of coefficients T(n,k) of x^(2n-1)*m^k in S(x,m) for n>=1, k=0..n-1.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 14, 14, 1, 1, 30, 81, 30, 1, 1, 55, 308, 308, 55, 1, 1, 91, 910, 1872, 910, 91, 1, 1, 140, 2268, 8250, 8250, 2268, 140, 1, 1, 204, 4998, 29172, 51425, 29172, 4998, 204, 1, 1, 285, 10032, 87780, 247247, 247247, 87780, 10032, 285, 1, 1, 385, 18711, 233376, 980980, 1565109, 980980, 233376, 18711, 385, 1, 1, 506, 32890, 562419, 3354780, 7970144, 7970144, 3354780, 562419, 32890, 506, 1

Offset: 1

Paul D. Hanna, Nov 29 2016

T(n,k) = the number of fighting fish with (n-k) left lower free and (k+1) right lower free edges with a marked tail. [See Theorem 3 in the Duchi reference on Fighting Fish: enumerative properties.] - Paul D. Hanna, Dec 08 2016

			This triangle of coefficients of x^(2*n-1)*m^k in S(x,m) for n>=1, k=0..n-1, begins:
  1;
  1, 1;
  1, 5, 1;
  1, 14, 14, 1;
  1, 30, 81, 30, 1;
  1, 55, 308, 308, 55, 1;
  1, 91, 910, 1872, 910, 91, 1;
  1, 140, 2268, 8250, 8250, 2268, 140, 1;
  1, 204, 4998, 29172, 51425, 29172, 4998, 204, 1;
  1, 285, 10032, 87780, 247247, 247247, 87780, 10032, 285, 1;
  1, 385, 18711, 233376, 980980, 1565109, 980980, 233376, 18711, 385, 1;
  1, 506, 32890, 562419, 3354780, 7970144, 7970144, 3354780, 562419, 32890, 506, 1; ...
Generating function:
S(x,m) = x + (m + 1)*x^3 + (m^2 + 5*m + 1)*x^5 +
 (m^3 + 14*m^2 + 14*m + 1)*x^7 +
 (m^4 + 30*m^3 + 81*m^2 + 30*m + 1)*x^9 +
 (m^5 + 55*m^4 + 308*m^3 + 308*m^2 + 55*m + 1)*x^11 +
 (m^6 + 91*m^5 + 910*m^4 + 1872*m^3 + 910*m^2 + 91*m + 1)*x^13 +
 (m^7 + 140*m^6 + 2268*m^5 + 8250*m^4 + 8250*m^3 + 2268*m^2 + 140*m + 1)*x^15 +...
where S = S(x,m) satisfies:
S = x / ( G(-S^2) * G(-m*S^2) ) such that G(x) = 1 + x*G(x)^2.
Also,
S = x * (1 + x*S) * (1 + m*x*S) / (1 - m*x^2*S^2)^2,
where related series C = C(x,m) and D = D(x,m) satisfy
S = x*C*D, C = 1 + x*S*D, and D = 1 + m*x*S*C,
such that
C = C^2 - S^2,
D = D^2 - m*S^2.
...
The square of the g.f. begins:
S(x,m)^2 = x^2 + (2*m + 2)*x^4 + (3*m^2 + 12*m + 3)*x^6 +
 (4*m^3 + 40*m^2 + 40*m + 4)*x^8 + (5*m^4 + 100*m^3 + 245*m^2 + 100*m + 5)*x^10 +
 (6*m^5 + 210*m^4 + 1008*m^3 + 1008*m^2 + 210*m + 6)*x^12 +
 (7*m^6 + 392*m^5 + 3234*m^4 + 6300*m^3 + 3234*m^2 + 392*m + 7)*x^14 +
 (8*m^7 + 672*m^6 + 8736*m^5 + 29040*m^4 + 29040*m^3 + 8736*m^2 + 672*m + 8)*x^16 +...+ x^(2*n)*Sum_{k=0,n-1} n*A082680(n,k+1)*m^k +...
where A082680(n,k+1) = (n+k)!*(2*n-k-1)!/((k+1)!*(n-k)!*(2*k+1)!*(2*n-2*k-1)!).

Paul D. Hanna, Table of n, a(n) for n = 1..1035 for rows 1..45 of the flattened form of this triangle.
Enrica Duchi, Veronica Guerrini, Simone Rinaldi, and Gilles Schaeffer, Fighting Fish: enumerative properties, arXiv:1611.04625 [math.CO], 2016.
Thomas Einolf, Robert Muth, and Jeffrey Wilkinson, Injectively k-colored rooted forests, arXiv:2107.13417 [math.CO], 2021.
Zai-Ting Huang, Congruence Properties From Fish Numbers, Master's thesis, Nat'l Taiwan Normal Univ. (2025) 31944792.

Cf. A278881 (C(x,m)), A278882 (D(x,m)), A278883 (central terms).

Cf. A000108, A006013 (row sums), A258313, A278745, A082680.

T[n_, k_] := (2n-1)/((2n-2k-1)(2k+1)) Binomial[2n-k-2, k] Binomial[n+k-1, n-k-1];
Table[T[n, k], {n, 1, 12}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Jul 26 2018 *)

{T(n,k) = my(S=x,C=1,D=1); for(i=0,2*n, S = x*C*D + O(x^(2*n+2)); C = 1 + x*S*D; D = 1 + m*x*S*C;); polcoeff(polcoeff(S,2*n-1,x),k,m)}
for(n=1,15, for(k=0,n-1, print1(T(n,k),", "));print(""))

/* Explicit formula for T(n,k) */
{T(n,k) = (2*n-1)/((2*n-2*k-1)*(2*k+1)) * binomial(2*n-k-2,k) * binomial(n+k-1,n-k-1)}
for(n=1,15, for(k=0,n-1, print1(T(n,k),", "));print(""))

G.f. S = S(x,m), and related functions C = C(x,m) and D = D(x,m) satisfy:
(1.a) S = x*C*D.
(1.b) C = 1 + x*S*D.
(1.c) D = 1 + m*x*S*C.
...
(2.a) C = C^2 - S^2.
(2.b) D = D^2 - m*S^2.
(2.c) C = (1 + sqrt(1 + 4*S^2))/2.
(2.d) D = (1 + sqrt(1 + 4*m*S^2))/2.
...
(3.a) S = x*(1 + x*S)*(1 + m*x*S) / (1 - m*x^2*S^2)^2.
(3.b) C = (1 + x*S) / (1 - m*x^2*S^2).
(3.c) D = (1 + m*x*S) / (1 - m*x^2*S^2).
(3.d) S = x/((1 - x^2*D^2)*(1 - m*x^2*C^2)).
(3.e) C = 1/(1 - x^2*D^2).
(3.f) D = 1/(1 - m*x^2*C^2).
...
(4.a) x = m^2*x^4*S^5 - 2*m*x^2*S^3 - m*x^3*S^2 + (1 - (m+1)*x^2)*S.
(4.b) 0 = 1 - (1-x^2)*C - 2*m*x^2*C^2 + 2*m*x^2*C^3 + m^2*x^4*C^4 - m^2*x^4*C^5.
(4.c) 0 = 1 - (1-m*x^2)*D - 2*x^2*D^2 + 2*x^2*D^3 + x^4*D^4 - x^4*D^5.
...
(5.a) S(x,m) = Series_Reversion( x*G(-x^2)*G(-m*x^2) ), where G(x) = 1 + x*G(x)^2 is the g.f. of the Catalan numbers (A000108).
Logarithmic derivatives.
(6.a) C'/C = 2*S*S' / (C^2 + S^2).
(6.b) D'/D = 2*m*S*S' / (D^2 + m*S^2).
...
(7.a) S(x,m)^2 = Sum_{n>=1} x^(2*n) * Sum_{k=0,n-1} n*A082680(n,k+1)*m^k, where A082680(n,k+1) = (n+k)!*(2*n-k-1)!/((k+1)!*(n-k)!*(2*k+1)!*(2*n-2*k-1)!).
...
T(n,k) = (2*n-1)/((2*n-2*k-1)*(2*k+1)) * binomial(2*n-k-2,k) * binomial(n+k-1,n-k-1). [From Theorem 3 in the Duchi reference] - Paul D. Hanna, Dec 08 2016
Row sums yield A006013(n-1) = binomial(3*n-2,n-1)/n for n>=1.
Central terms: T(2*n+1, n) = (4*n-3) * ( binomial(3*n-3,n-1)/(2*n-1) )^2 for n>=1.
Sum_{k=0..n-1} 2^k * T(n,k) = A258313(n-1) for n>=1.
Sum_{k=0..2*n-2} (-1)^k * T(2*n-1,k) = A278745(n) for n>=1.

A342989 Triangle read by rows: T(n,k) is the number of nonseparable rooted toroidal maps with n edges and k faces, n >= 2, k = 1..n-1.

Original entry on oeis.org

1, 4, 4, 10, 39, 10, 20, 190, 190, 20, 35, 651, 1568, 651, 35, 56, 1792, 8344, 8344, 1792, 56, 84, 4242, 33580, 64667, 33580, 4242, 84, 120, 8988, 111100, 361884, 361884, 111100, 8988, 120, 165, 17490, 317680, 1607125, 2713561, 1607125, 317680, 17490, 165

Offset: 2

Andrew Howroyd, Apr 04 2021

The number of vertices is n - k.

Column k is a polynomial of degree 3*k. 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;
    4,    4;
   10,   39,     10;
   20,  190,    190,     20;
   35,  651,   1568,    651,     35;
   56, 1792,   8344,   8344,   1792,     56;
   84, 4242,  33580,  64667,  33580,   4242,   84;
  120, 8988, 111100, 361884, 361884, 111100, 8988, 120;
  ...

Andrew Howroyd, Table of n, a(n) for n = 2..1276 (first 50 rows)
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 II.

Columns 1..4 are A000292, A006408, A006409, A006410.
Row sums are A343089.
Cf. A082680 (planar case), A269921 (rooted toroidal maps), A343090, A343092.

MQ(n,g,x=1)={ \\ after Quadric in A269921.
  my(Q=matrix(n+1,g+1)); Q[1,1]=x;
  for(n=1, n, for(g=0, min(n\2,g),
     my(t = (1+x)*(2*n-1)/3 * Q[n, 1+g]
       + if(g && n>1, (2*n-3)*(2*n-2)*(2*n-1)/12 * Q[n-1, g])
       + sum(k = 1, n-1, sum(i = 0, g, (2*k-1) * (2*(n-k)-1) * Q[k, 1+i] * Q[n-k, 1+g-i]))/2);
     Q[1+n, 1+g] = t * 6/(n+1); ));
  Q
}
F(n,m,y,z)={my(Q=MQ(n,m,z)); sum(n=0, n, x^n*Ser(Q[1+n,]/z, y)) + O(x*x^n)}
H(n,g=1)={my(p=F(n,g,'y,'z), v=Vec(polcoef(subst(p, x, serreverse(x*p^2)), g, 'y))); vector(#v, n, Vecrev(v[n], n))}
{ my(T=H(10)); for(n=1, #T, print(T[n])) }

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

A278881 Triangle where g.f. C = C(x,m) and related series S = S(x,m) and D = D(x,m) satisfy S = xCD, C = 1 + xSD, and D = 1 + mxS*C, as read by rows of coefficients T(n,k) of x^(2n)m^k in C(x,m) for n>=0, k=0..n.

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 8, 3, 0, 1, 20, 30, 4, 0, 1, 40, 147, 80, 5, 0, 1, 70, 504, 672, 175, 6, 0, 1, 112, 1386, 3600, 2310, 336, 7, 0, 1, 168, 3276, 14520, 18150, 6552, 588, 8, 0, 1, 240, 6930, 48048, 102245, 72072, 16170, 960, 9, 0, 1, 330, 13464, 137280, 455455, 546546, 240240, 35904, 1485, 10, 0, 1, 440, 24453, 350064, 1701700, 3179904, 2382380, 700128, 73359, 2200, 11, 0, 1, 572, 42042, 815100, 5542680, 15148224, 17672928, 8868288, 1833975, 140140, 3146, 12, 0

Offset: 0

Paul D. Hanna, Nov 29 2016

			This triangle of coefficients of x^(2*n)*m^k in C(x,m) for n>=0, k=0..n, begins:
1;
1, 0;
1, 2, 0;
1, 8, 3, 0;
1, 20, 30, 4, 0;
1, 40, 147, 80, 5, 0;
1, 70, 504, 672, 175, 6, 0;
1, 112, 1386, 3600, 2310, 336, 7, 0;
1, 168, 3276, 14520, 18150, 6552, 588, 8, 0;
1, 240, 6930, 48048, 102245, 72072, 16170, 960, 9, 0;
1, 330, 13464, 137280, 455455, 546546, 240240, 35904, 1485, 10, 0;
1, 440, 24453, 350064, 1701700, 3179904, 2382380, 700128, 73359, 2200, 11, 0;
1, 572, 42042, 815100, 5542680, 15148224, 17672928, 8868288, 1833975, 140140, 3146, 12, 0; ...
Generating function:
C(x,m) = 1 + x^2 + (1 + 2*m)*x^4 + (1 + 8*m + 3*m^2)*x^6 +
(1 + 20*m + 30*m^2 + 4*m^3)*x^8 +
(1 + 40*m + 147*m^2 + 80*m^3 + 5*m^4)*x^10 +
(1 + 70*m + 504*m^2 + 672*m^3 + 175*m^4 + 6*m^5)*x^12 +
(1 + 112*m + 1386*m^2 + 3600*m^3 + 2310*m^4 + 336*m^5 + 7*m^6)*x^14 +
(1 + 168*m + 3276*m^2 + 14520*m^3 + 18150*m^4 + 6552*m^5 + 588*m^6 + 8*m^7)*x^16 +...
where g.f. C = C(x,m) and related series S = S(x,m) and D = D(x,m) satisfy
S = x*C*D, C = 1 + x*S*D, and D = 1 + m*x*S*C,
such that
C = C^2 - S^2,
D = D^2 - m*S^2.
The square of the g.f. begins:
C(x,m)^2 = 1 + 2*x^2 + (4*m + 3)*x^4 + (6*m^2 + 20*m + 4)*x^6 +
(8*m^3 + 70*m^2 + 60*m + 5)*x^8 +
(10*m^4 + 180*m^3 + 392*m^2 + 140*m + 6)*x^10 +
(12*m^5 + 385*m^4 + 1680*m^3 + 1512*m^2 + 280*m + 7)*x^12 +
(14*m^6 + 728*m^5 + 5544*m^4 + 9900*m^3 + 4620*m^2 + 504*m + 8)*x^14 +
(16*m^7 + 1260*m^6 + 15288*m^5 + 47190*m^4 + 43560*m^3 + 12012*m^2 + 840*m + 9)*x^16 +
(18*m^8 + 2040*m^7 + 36960*m^6 + 180180*m^5 + 286286*m^4 + 156156*m^3 + 27720*m^2 + 1320*m + 10)*x^18 +...

Paul D. Hanna, Table of n, a(n) for n = 0..1080 for rows 0..45 of the flattened form of this triangle.
Thomas Einolf, Robert Muth, and Jeffrey Wilkinson, Injectively k-colored rooted forests, arXiv:2107.13417 [math.CO], 2021.
Tad White, Quota Trees, arXiv:2401.01462 [math.CO], 2024. See p. 20.

Cf. A278880 (S(x,m)), A278882 (D(x,m)), A278884 (central terms).

Cf. A001764 (row sums), A000108, A258314 (C(x,m) at m=2), A243863.

{T(n,k) = my(S=x,C=1,D=1); for(i=0,2*n, S = x*C*D + O(x^(2*n+2)); C = 1 + x*S*D; D = 1 + m*x*S*C;); polcoeff(polcoeff(C,2*n,x),k,m)}
for(n=0,15, for(k=0,n, print1(T(n,k),", "));print(""))

/* Explicit formula for T(n, k) */
{T(n,k) = if(k==0,1, if(n==k,0, (n+k)!*(2*n-k-1)!/(k!*(n-k)!*(2*k+1)!*(2*n-2*k-1)!) ))}
for(n=0, 15, for(k=0, n, print1(T(n, k), ", ")); print("")) \\ Paul D. Hanna, Dec 11 2016

G.f. C = C(x,m), and related functions S = S(x,m) and D = D(x,m) satisfy:
(1.a) S = x*C*D.
(1.b) C = 1 + x*S*D.
(1.c) D = 1 + m*x*S*C.
...
(2.a) C = C^2 - S^2.
(2.b) D = D^2 - m*S^2.
(2.c) C = (1 + sqrt(1 + 4*S^2))/2.
(2.d) D = (1 + sqrt(1 + 4*m*S^2))/2.
...
(3.a) S = x*(1 + x*S)*(1 + m*x*S) / (1 - m*x^2*S^2)^2.
(3.b) C = (1 + x*S) / (1 - m*x^2*S^2).
(3.c) D = (1 + m*x*S) / (1 - m*x^2*S^2).
(3.d) S = x/((1 - x^2*D^2)*(1 - m*x^2*C^2)).
(3.e) C = 1/(1 - x^2*D^2).
(3.f) D = 1/(1 - m*x^2*C^2).
...
(4.a) x = m^2*x^4*S^5 - 2*m*x^2*S^3 - m*x^3*S^2 + (1 - (m+1)*x^2)*S.
(4.b) 0 = 1 - (1-x^2)*C - 2*m*x^2*C^2 + 2*m*x^2*C^3 + m^2*x^4*C^4 - m^2*x^4*C^5.
(4.c) 0 = 1 - (1-m*x^2)*D - 2*x^2*D^2 + 2*x^2*D^3 + x^4*D^4 - x^4*D^5.
...
(5.a) S(x,m) = Series_Reversion( x*G(-x^2)*G(-m*x^2) ), where G(x) = 1 + x*G(x)^2 is the g.f. of the Catalan numbers (A000108).
Logarithmic derivatives.
(6.a) C'/C = 2*S*S' / (C^2 + S^2).
(6.b) D'/D = 2*m*S*S' / (D^2 + m*S^2).
...
T(n,k) = (k+1) * A082680(n+1,k+1) for n>=0 with T(0,0) = 1 and T(n,n) = 1 for n>0. - Paul D. Hanna, Dec 11 2016
T(n,k) = (n+k)!*(2*n-k-1)!/(k!*(n-k)!*(2*k+1)!*(2*n-2*k-1)!) for n>k>0 with T(n,0) = 1 and T(n,n) = 0 for n>0. - Paul D. Hanna, Dec 11 2016
Row sums yield A001764(n) = binomial(3*n,n)/(2*n+1).
Central terms: T(2*n,n) = binomial(3*n-1,n) * binomial(3*n,n)/(2*n+1).
Sum_{k=0..n} 2^k * T(n,k) = A258314(n-1) for n>=0.
Sum_{k=0..n} (-1)^k * T(n,k) = A243863(n) for n>=0.

A278882 Triangle where g.f. D = D(x,m) and related series S = S(x,m) and C = C(x,m) satisfy S = xCD, C = 1 + xSD, and D = 1 + mxS*C, as read by rows of coefficients T(n,k) of x^(2n)m^k in C(x,m) for n>=1, k=0..n.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 3, 8, 1, 0, 4, 30, 20, 1, 0, 5, 80, 147, 40, 1, 0, 6, 175, 672, 504, 70, 1, 0, 7, 336, 2310, 3600, 1386, 112, 1, 0, 8, 588, 6552, 18150, 14520, 3276, 168, 1, 0, 9, 960, 16170, 72072, 102245, 48048, 6930, 240, 1, 0, 10, 1485, 35904, 240240, 546546, 455455, 137280, 13464, 330, 1, 0, 11, 2200, 73359, 700128, 2382380, 3179904, 1701700, 350064, 24453, 440, 1, 0, 12, 3146, 140140, 1833975, 8868288, 17672928, 15148224, 5542680, 815100, 42042, 572, 1

Offset: 0

Paul D. Hanna, Nov 29 2016

			This triangle of coefficients of x^(2*n)*m^k in D(x,m) for n>=0, k=0..n, begins:
1;
0, 1;
0, 2, 1;
0, 3, 8, 1;
0, 4, 30, 20, 1;
0, 5, 80, 147, 40, 1;
0, 6, 175, 672, 504, 70, 1;
0, 7, 336, 2310, 3600, 1386, 112, 1;
0, 8, 588, 6552, 18150, 14520, 3276, 168, 1;
0, 9, 960, 16170, 72072, 102245, 48048, 6930, 240, 1;
0, 10, 1485, 35904, 240240, 546546, 455455, 137280, 13464, 330, 1;
0, 11, 2200, 73359, 700128, 2382380, 3179904, 1701700, 350064, 24453, 440, 1;
0, 12, 3146, 140140, 1833975, 8868288, 17672928, 15148224, 5542680, 815100, 42042, 572, 1; ...
Generating function:
D(x,m) = 1 + m*x^2 + (2*m + m^2)*x^4 + (3*m + 8*m^2 + m^3)*x^6 +
(4*m + 30*m^2 + 20*m^3 + m^4)*x^8 +
(5*m + 80*m^2 + 147*m^3 + 40*m^4 + m^5)*x^10 +
(6*m + 175*m^2 + 672*m^3 + 504*m^4 + 70*m^5 + m^6)*x^12 +
(7*m + 336*m^2 + 2310*m^3 + 3600*m^4 + 1386*m^5 + 112*m^6 + m^7)*x^14 +
(8*m + 588*m^2 + 6552*m^3 + 18150*m^4 + 14520*m^5 + 3276*m^6 + 168*m^7 + m^8)*x^16 +...
where g.f. D = D(x,m) and related series S = S(x,m) and C = C(x,m) satisfy
S = x*C*D, C = 1 + x*S*D, and D = 1 + m*x*S*C,
such that
C = C^2 - S^2,
D = D^2 - m*S^2.
The square of the g.f. begins:
D(x,m)^2 = 1 + 2*m*x^2 + (3*m^2 + 4*m)*x^4 +
(4*m^3 + 20*m^2 + 6*m)*x^6 +
(5*m^4 + 60*m^3 + 70*m^2 + 8*m)*x^8 +
(6*m^5 + 140*m^4 + 392*m^3 + 180*m^2 + 10*m)*x^10 +
(7*m^6 + 280*m^5 + 1512*m^4 + 1680*m^3 + 385*m^2 + 12*m)*x^12 +
(8*m^7 + 504*m^6 + 4620*m^5 + 9900*m^4 + 5544*m^3 + 728*m^2 + 14*m)*x^14 +
(9*m^8 + 840*m^7 + 12012*m^6 + 43560*m^5 + 47190*m^4 + 15288*m^3 + 1260*m^2 + 16*m)*x^16 +
(10*m^9 + 1320*m^8 + 27720*m^7 + 156156*m^6 + 286286*m^5 + 180180*m^4 + 36960*m^3 + 2040*m^2 + 18*m)*x^18 +...

Paul D. Hanna, Table of n, a(n) for n = 0..1080 for rows 0..45 of the flattened form of this triangle.

Cf. A278880 (S(x,m)), A278881 (C(x,m)), A278884 (central terms).

Cf. A001764 (row sums), A000108, A258315, A243863.

{T(n,k) = my(S=x,C=1,D=1); for(i=0,2*n, S = x*C*D + O(x^(2*n+2)); C = 1 + x*S*D; D = 1 + m*x*S*C;); polcoeff(polcoeff(D,2*n,x),k,m)}
for(n=0,15, for(k=0,n, print1(T(n,k),", "));print(""))

/* Explicit formula for T(n, k) */
{T(n, k) = if(n==k, 1, if(k==0, 0, (2*n-k)!*(n+k-1)!/(k!*(n-k)!*(2*k-1)!*(2*n-2*k+1)!) ))}
for(n=0, 15, for(k=0, n, print1(T(n, k), ", ")); print("")) \\ Paul D. Hanna, Dec 11 2016

G.f. D = D(x,m), and related functions S = S(x,m) and C = C(x,m) satisfy:
(1.a) S = x*C*D.
(1.b) C = 1 + x*S*D.
(1.c) D = 1 + m*x*S*C.
...
(2.a) C = C^2 - S^2.
(2.b) D = D^2 - m*S^2.
(2.c) C = (1 + sqrt(1 + 4*S^2))/2.
(2.d) D = (1 + sqrt(1 + 4*m*S^2))/2.
...
(3.a) S = x*(1 + x*S)*(1 + m*x*S) / (1 - m*x^2*S^2)^2.
(3.b) C = (1 + x*S) / (1 - m*x^2*S^2).
(3.c) D = (1 + m*x*S) / (1 - m*x^2*S^2).
(3.d) S = x/((1 - x^2*D^2)*(1 - m*x^2*C^2)).
(3.e) C = 1/(1 - x^2*D^2).
(3.f) D = 1/(1 - m*x^2*C^2).
...
(4.a) x = m^2*x^4*S^5 - 2*m*x^2*S^3 - m*x^3*S^2 + (1 - (m+1)*x^2)*S.
(4.b) 0 = 1 - (1-x^2)*C - 2*m*x^2*C^2 + 2*m*x^2*C^3 + m^2*x^4*C^4 - m^2*x^4*C^5.
(4.c) 0 = 1 - (1-m*x^2)*D - 2*x^2*D^2 + 2*x^2*D^3 + x^4*D^4 - x^4*D^5.
...
(5.a) S(x,m) = Series_Reversion( x*G(-x^2)*G(-m*x^2) ), where G(x) = 1 + x*G(x)^2 is the g.f. of the Catalan numbers (A000108).
Logarithmic derivatives.
(6.a) C'/C = 2*S*S' / (C^2 + S^2).
(6.b) D'/D = 2*m*S*S' / (D^2 + m*S^2).
...
T(n,k) = (n-k+1) * A082680(n+1,n-k+1) for n>=0 with T(0,0) = 1 and T(n,0) = 0 for n>0. - Paul D. Hanna, Dec 11 2016
T(n,k) = (2*n-k)!*(n+k-1)!/(k!*(n-k)!*(2*k-1)!*(2*n-2*k+1)!) for n>k>0 with T(n,0) = 1 and T(n,n) = 0 for n>0. - Paul D. Hanna, Dec 11 2016
Row sums yield A001764(n) = binomial(3*n,n)/(2*n+1).
Central terms: T(2*n,n) = binomial(3*n-1,n) * binomial(3*n,n)/(2*n+1).
Sum_{k=0..n} 2^k * T(n,k) = A258315(n-1) for n>=0.
Sum_{k=0..n} (-1)^k * T(n,k) = (-1)^n * A243863(n) for n>=0.

A342061 Triangle read by rows: T(n,k) is the number of sensed 2-connected (nonseparable) planar maps with n edges and k vertices, n >= 2, 2 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 8, 3, 1, 1, 4, 16, 16, 4, 1, 1, 5, 38, 63, 38, 5, 1, 1, 7, 72, 218, 218, 72, 7, 1, 1, 8, 134, 622, 1075, 622, 134, 8, 1, 1, 10, 224, 1600, 4214, 4214, 1600, 224, 10, 1, 1, 12, 375, 3703, 14381, 22222, 14381, 3703, 375, 12, 1

Offset: 2

Andrew Howroyd, Mar 30 2021

The number of faces is n + 2 - k.

			Triangle begins:
  1;
  1, 1;
  1, 1,   1;
  1, 2,   2,   1;
  1, 3,   8,   3,    1;
  1, 4,  16,  16,    4,   1;
  1, 5,  38,  63,   38,   5,   1;
  1, 7,  72, 218,  218,  72,   7, 1;
  1, 8, 134, 622, 1075, 622, 134, 8, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 2..1276 (first 50 rows)
Timothy R. Walsh, Efficient enumeration of sensed planar maps, Discrete Math. 293 (2005), no. 1-3, 263--289. MR2136069 (2006b:05062).

Column k=3 is A001399(n-3).
Row sums are A006402.
Cf. A082680 (rooted), A239893, A342059.

\\ See section 4 of Walsh reference.
T(n)={
  my(B=matrix(n, n, i, j, if(i+j <= n+1, (2*i+j-2)!*(2*j+i-2)!/(i!*j!*(2*i-1)!*(2*j-1)!))));
  my(C(i,j)=((i+j-1)*(i+1)*(j+1)/(2*(2*i+j-1)*(2*j+i-1)))*B[(i+1)/2,(j+1)/2]);
  my(D(i,j)=((j+1)/2)*B[i/2, (j+1)/2]);
  my(E(i,j)=((i-1)*(j-1) + 2*(i+j)*(i+j-1))*B[i,j]);
  my(F(i,j)=if(!i, j==1, ((i+j)*(6*j+2*i-5)*j*(2*i+j-1)/(2*(2*i+1)*(2*j+i-2)))*B[i,j]) + if(j-1, binomial(i+2,2)*B[i+1,j-1]));
  vector(n, n, vector(n, i, my(j=n+1-i); B[i,j]
    + (i+j)*if(i%2, if(j%2, C(i,j), D(j,i)), if(j%2, D(i,j)))
    + sumdiv(i+j, d, if(d>1, eulerphi(d)*( if(i%d==0, E(i/d, j/d) ) + if(i%d==1, F((i-1)/d, (j+1)/d)) + if(j%d==1, F((j-1)/d, (i+1)/d)) )))
   )/(2*n+2));
}
{ my(A=T(10)); for(n=1, #A, print(A[n])) }

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

A379432 Triangle read by rows: T(n,k) is the number of unsensed 2-connected (nonseparable) planar maps with n edges and k vertices, n >= 2, 2 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 7, 3, 1, 1, 4, 13, 13, 4, 1, 1, 5, 29, 44, 29, 5, 1, 1, 7, 51, 139, 139, 51, 7, 1, 1, 8, 92, 370, 623, 370, 92, 8, 1, 1, 10, 147, 913, 2307, 2307, 913, 147, 10, 1, 1, 12, 240, 2048, 7644, 11673, 7644, 2048, 240, 12, 1, 1, 14, 357, 4295, 22344, 50174, 50174, 22344, 4295, 357, 14, 1

Offset: 2

Andrew Howroyd, Jan 14 2025

The maps considered here may include parallel edges.

The number of faces is n + 2 - k.

			Triangle begins:
   1;
   1,  1;
   1,  1,   1;
   1,  2,   2,   1;
   1,  3,   7,   3,    1;
   1,  4,  13,  13,    4,    1;
   1,  5,  29,  44,   29,    5,   1;
   1,  7,  51, 139,  139,   51,   7,   1;
   1,  8,  92, 370,  623,  370,  92,   8,  1;
   1, 10, 147, 913, 2307, 2307, 913, 147, 10, 1;
   ...

Timothy R. Walsh, Number of sensed planar maps with n edges and m vertices, pp. 29-36.

Row sums are A006403.

Cf. A082680 (rooted), A342061 (sensed), A212438 (3-connected), A277741, A342060.

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

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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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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A278880 Triangle where g.f. S = S(x,m) satisfies: S = x/(G(-S^2)*G(-m*S^2)) such that G(x) = 1 + x*G(x)^2 is the g.f. of the Catalan numbers (A000108), as read by rows of coefficients T(n,k) of x^(2*n-1)*m^k in S(x,m) for n>=1, k=0..n-1.

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A342989 Triangle read by rows: T(n,k) is the number of nonseparable rooted toroidal maps with n edges and k faces, n >= 2, k = 1..n-1.

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A278881 Triangle where g.f. C = C(x,m) and related series S = S(x,m) and D = D(x,m) satisfy S = x*C*D, C = 1 + x*S*D, and D = 1 + m*x*S*C, as read by rows of coefficients T(n,k) of x^(2*n)*m^k in C(x,m) for n>=0, k=0..n.

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A278882 Triangle where g.f. D = D(x,m) and related series S = S(x,m) and C = C(x,m) satisfy S = x*C*D, C = 1 + x*S*D, and D = 1 + m*x*S*C, as read by rows of coefficients T(n,k) of x^(2*n)*m^k in C(x,m) for n>=1, k=0..n.

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A342061 Triangle read by rows: T(n,k) is the number of sensed 2-connected (nonseparable) planar maps with n edges and k vertices, n >= 2, 2 <= k <= n.

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A278880 Triangle where g.f. S = S(x,m) satisfies: S = x/(G(-S^2)G(-mS^2)) such that G(x) = 1 + xG(x)^2 is the g.f. of the Catalan numbers (A000108), as read by rows of coefficients T(n,k) of x^(2n-1)*m^k in S(x,m) for n>=1, k=0..n-1.

A278881 Triangle where g.f. C = C(x,m) and related series S = S(x,m) and D = D(x,m) satisfy S = xCD, C = 1 + xSD, and D = 1 + mxS*C, as read by rows of coefficients T(n,k) of x^(2n)m^k in C(x,m) for n>=0, k=0..n.

A278882 Triangle where g.f. D = D(x,m) and related series S = S(x,m) and C = C(x,m) satisfy S = xCD, C = 1 + xSD, and D = 1 + mxS*C, as read by rows of coefficients T(n,k) of x^(2n)m^k in C(x,m) for n>=1, k=0..n.