A210252 Triangle read by rows: T(n,k) is the number of c-nets with n+1 faces and k+1 vertices, 1 <= k <= n. But see A290326 for a better version.

0, 0, 0, 0, 0, 1, 0, 0, 0, 4, 0, 0, 0, 3, 24, 0, 0, 0, 0, 33, 188, 0, 0, 0, 0, 13, 338, 1705, 0, 0, 0, 0, 0, 252, 3580, 16980, 0, 0, 0, 0, 0, 68, 3740, 39525, 180670, 0, 0, 0, 0, 0, 0, 1938, 51300, 452865, 2020120, 0, 0, 0, 0, 0, 0, 399, 38076, 685419, 5354832, 23478426, 0, 0, 0, 0, 0, 0, 0, 15180, 646415, 9095856, 65022840, 281481880, 0, 0, 0, 0, 0, 0, 0, 2530, 373175, 10215450, 120872850, 807560625, 3461873536, 0, 0, 0, 0, 0, 0, 0, 0, 121095, 7580040, 155282400, 1614234960, 10224817515, 43494961404

Offset: 1

Author

N. J. A. Sloane, Mar 19 2012

Keywords

• nonn
• tabl

Comments

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

This triangle is based on a mis-reading of the Mullin-Schellenberg table. See A290326 for a better version. - N. J. A. Sloane, Jul 28 2017

Examples

			Triangle begins:
n\k
[1]  0
[2]  0 0
[3]  0 0 1
[4]  0 0 0 4
[5]  0 0 0 3 24
[6]  0 0 0 0 33 188
[7]  0 0 0 0 13 338 1705
[8]  0 0 0 0 0 252 3580 16980
[9]  0 0 0 0 0 68 3740 39525 180670
[10] 0 0 0 0 0 0 1938 51300 452865 2020120
[11] 0 0 0 0 0 0 399 38076 685419 5354832 23478426
[12] 0 0 0 0 0 0 0 15180 646415 9095856 65022840 281481880
[13] 0 0 0 0 0 0 0 2530 373175 10215450 120872850 807560625 3461873536
[14] 0 0 0 0 0 0 0 0 121095 7580040 155282400 1614234960 10224817515 43494961404
...
		

Links

• Gheorghe Coserea, Rows n = 1..100, 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

Right-hand edge is A001506.

See A290326 for a better version.

Programs

PARI

T(n,m) = {
  if (m <= 1+n\2 || n < 3, return(0));
  sum(k=0, m-1, sum(j=0, n-1,
     (-1)^((k+j+1)%2) * binomial(k+j,k)*(k+j+1)*(k+j+2)/2*
     (binomial(2*n, m-k-1) * binomial(2*m, n-j-1) -
      4 * binomial(2*n-1, m-k-2) * binomial(2*m-1, n-j-2))));
};
concat(vector(14, n, vector(n,m, T(n,m))))  \\ Gheorghe Coserea, Jan 08 2017

Formula

T(n,m) = Sum_{k=0..m-1} Sum_{j=0..n-1} (-1)^(k+j+1) * ((k+j+2)!/(2!*k!*j!)) * (binomial(2*n, m-k-1) * binomial(2*m, n-j-1) - 4 * binomial(2*n-1, m-k-2) * binomial(2*m-1, n-j-2)) if (n+2)/2 < m <= n and 0 otherwise. - Sean A. Irvine, Sep 28 2015

Extensions

a(105)=T(14,14) corrected by Sean A. Irvine, Sep 28 2015

Name changed by Gheorghe Coserea, Jul 23 2017