A290326 Triangle read by rows: T(n,k) is the number of c-nets with n+1 faces and k+1 vertices.

0, 0, 0, 0, 0, 1, 0, 0, 0, 4, 3, 0, 0, 0, 3, 24, 33, 13, 0, 0, 0, 0, 33, 188, 338, 252, 68, 0, 0, 0, 0, 13, 338, 1705, 3580, 3740, 1938, 399, 0, 0, 0, 0, 0, 252, 3580, 16980, 39525, 51300, 38076, 15180, 2530, 0, 0, 0, 0, 0, 68, 3740, 39525, 180670, 452865, 685419, 646415, 373175, 121095, 16965, 0, 0, 0, 0, 0, 0, 1938, 51300, 452865, 2020120, 5354832, 9095856, 10215450, 7580040, 3585270, 981708, 118668

Offset: 1

Author

Gheorghe Coserea, Jul 27 2017

Keywords

• nonn
• tabf

Comments

Row n >= 3 contains 2*n-3 terms.

c-nets are 3-connected rooted planar maps. This array also counts simple triangulations.

Table in Mullin & Schellenberg has incorrect values T(14,14) = 43494961412, T(15,13) = 21697730849, T(15,14) = 131631305614, T(15,15) = 556461655783. - Sean A. Irvine, Sep 28 2015

Examples

			A(x;t) = t^3*x^3 + (4*t^4 + 3*t^5)*x^4 + (3*t^4 + 24*t^5 + 33*t^6 + 13*t^7)*x^5 + ...
Triangle starts:
n\k  [1] [2] [3] [4] [5] [6]  [7]   [8]    [9]    [10]   [11]   [12]   [13]
[1]  0;
[2]  0,  0;
[3]  0,  0,  1;
[4]  0,  0,  0,  4,  3;
[5]  0,  0,  0,  3,  24, 33,  13;
[6]  0,  0,  0,  0,  33, 188, 338,  252,   68;
[7]  0,  0,  0,  0,  13, 338, 1705, 3580,  3740,  1938,  399;
[8]  0,  0,  0,  0,  0,  252, 3580, 16980, 39525, 51300, 38076, 15180, 2530;
[9]  ...
		

Links

• Gheorghe Coserea, Rows n = 1..103, flattened
• R. C. Mullin, P. J. Schellenberg, The enumeration of c-nets via quadrangulations, J. Combinatorial Theory 4 1968 259--276. MR0218275 (36 #1362).

Crossrefs

Cf. A000260, A001506, A001507, A001508.

Rows/Columns sum give A106651 (enumeration of c-nets by the number of vertices).

Antidiagonals sum give A000287 (enumeration of c-nets by the number of edges).

Programs

PARI

T(n,k) = {
  if (n < 3 || k < 3, return(0));
  sum(i=0, k-1, sum(j=0, n-1,
     (-1)^((i+j+1)%2) * binomial(i+j, i)*(i+j+1)*(i+j+2)/2*
     (binomial(2*n, k-i-1) * binomial(2*k, n-j-1) -
      4 * binomial(2*n-1, k-i-2) * binomial(2*k-1, n-j-2))));
};
N=10; concat(concat([0,0,0], apply(n->vector(2*n-3, k, T(n,k)), [3..N])))
\\ test 1: N=100; y=x*Ser(vector(N, n, sum(i=1+(n+2)\3, (2*n)\3-1, T(i,n-i)))); 0 == x*(x+1)^2*(x+2)*(4*x-1)*y' + 2*(x^2-11*x+1)*(x+1)^2*y + 10*x^6
/*
\\ test 2:
x='x; t='t; N=44; y=Ser(apply(n->Polrev(vector(2*n-3, k, T(n, k)), 't), [3..N+2]), 'x) * t*x^3;
0 == (t + 1)^3*(x + 1)^3*(t + x + t*x)^3*y^4 + t*(t + 1)^2*x*(x + 1)^2*((4*t^4 + 12*t^3 + 12*t^2 + 4*t)*x^4 + (12*t^4 + 16*t^3 - 4*t^2 - 8*t)*x^3 + (12*t^4 - 4*t^3 - 49*t^2 - 30*t + 3)*x^2 + (4*t^4 - 8*t^3 - 30*t^2 - 21*t)*x + 3*t^2)*y^3 + t^2*(t + 1)*x^2*(x + 1)*((6*t^5 + 18*t^4 + 18*t^3 + 6*t^2)*x^5 + (18*t^5 + 12*t^4 - 30*t^3 - 24*t^2)*x^4 + (18*t^5 - 30*t^4 - 123*t^3 - 58*t^2 + 17*t)*x^3 + (6*t^5 - 24*t^4 - 58*t^3 + 25*t^2 + 56*t)*x^2 + (17*t^3 + 56*t^2 + 48*t + 3)*x + 3*t)*y^2 + t^3*x^3*((4*t^6 + 12*t^5 + 12*t^4 + 4*t^3)*x^6 + (12*t^6 - 36*t^4 - 24*t^3)*x^5 + (12*t^6 - 36*t^5 - 99*t^4 - 26*t^3 + 25*t^2)*x^4 + (4*t^6 - 24*t^5 - 26*t^4 + 81*t^3 + 80*t^2)*x^3 + (25*t^4 + 80*t^3 + 44*t^2 - 14*t)*x^2 + (-14*t^2 - 17*t)*x + 1)*y + t^6*x^6*((t^4 + 2*t^3 + t^2)*x^4 + (2*t^4 - 7*t^3 - 9*t^2)*x^3 + (t^4 - 9*t^3 + 11*t)*x^2 + (11*t^2 + 13*t)*x - 1)
*/

Formula

T(n,k) = Sum_{i=0..k-1} Sum_{j=0..n-1} (-1)^(i+j+1) * ((i+j+2)!/(2!*i!*j!)) * (binomial(2*n, k-i-1) * binomial(2*k, n-j-1) - 4 * binomial(2*n-1, k-i-2) * binomial(2*k-1, n-j-2)) for all n >= 3, k >= 3.

A106651(n+1) = Sum_{k=1..2*n-3} T(n,k) for n >= 3.

A000287(n) = Sum_{i=1+floor((n+2)/3)..floor(2*n/3)-1} T(i,n-i).

A001506(n) = T(n,n), A001507(n) = T(n+1,n), A001508(n) = T(n+2,n).

A000260(n-2) = T(n, 2*n-3) for n>=3.

G.f. y = A(x;t) satisfies: 0 = (t + 1)^3*(x + 1)^3*(t + x + t*x)^3*y^4 + t*(t + 1)^2*x*(x + 1)^2*((4*t^4 + 12*t^3 + 12*t^2 + 4*t)*x^4 + (12*t^4 + 16*t^3 - 4*t^2 - 8*t)*x^3 + (12*t^4 - 4*t^3 - 49*t^2 - 30*t + 3)*x^2 + (4*t^4 - 8*t^3 - 30*t^2 - 21*t)*x + 3*t^2)*y^3 + t^2*(t + 1)*x^2*(x + 1)*((6*t^5 + 18*t^4 + 18*t^3 + 6*t^2)*x^5 + (18*t^5 + 12*t^4 - 30*t^3 - 24*t^2)*x^4 + (18*t^5 - 30*t^4 - 123*t^3 - 58*t^2 + 17*t)*x^3 + (6*t^5 - 24*t^4 - 58*t^3 + 25*t^2 + 56*t)*x^2 + (17*t^3 + 56*t^2 + 48*t + 3)*x + 3*t)*y^2 + t^3*x^3*((4*t^6 + 12*t^5 + 12*t^4 + 4*t^3)*x^6 + (12*t^6 - 36*t^4 - 24*t^3)*x^5 + (12*t^6 - 36*t^5 - 99*t^4 - 26*t^3 + 25*t^2)*x^4 + (4*t^6 - 24*t^5 - 26*t^4 + 81*t^3 + 80*t^2)*x^3 + (25*t^4 + 80*t^3 + 44*t^2 - 14*t)*x^2 + (-14*t^2 - 17*t)*x + 1)*y + t^6*x^6*((t^4 + 2*t^3 + t^2)*x^4 + (2*t^4 - 7*t^3 - 9*t^2)*x^3 + (t^4 - 9*t^3 + 11*t)*x^2 + (11*t^2 + 13*t)*x - 1). - Gheorghe Coserea, Sep 29 2018