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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.
1
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
1,10
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
Right-hand edge is A001506.
See A290326 for a better version.
Sequence in context: A368661 A369009 A345403 * A028719 A028662 A028715
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