OFFSET
1,10
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
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).
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
EXAMPLE
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
...
PROG
(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
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
N. J. A. Sloane, Mar 19 2012
EXTENSIONS
a(105)=T(14,14) corrected by Sean A. Irvine, Sep 28 2015
Name changed by Gheorghe Coserea, Jul 23 2017
STATUS
approved