A210252 - OEIS

%I #43 Jul 28 2017 08:55:56

%S 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,

%T 0,0,0,0,252,3580,16980,0,0,0,0,0,68,3740,39525,180670,0,0,0,0,0,0,

%U 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

%N 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.

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

%C 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

%C 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

%H Gheorghe Coserea, <a href="/A210252/b210252.txt">Rows n = 1..100, flattened</a>

%H R. C. Mullin, P. J. Schellenberg, <a href="http://dx.doi.org/10.1016/S0021-9800(68)80007-9">The enumeration of c-nets via quadrangulations</a>, J. Combinatorial Theory 4 1968 259--276. MR0218275 (36 #1362).

%F 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

%e Triangle begins:

%e n\k

%e [1]  0

%e [2]  0 0

%e [3]  0 0 1

%e [4]  0 0 0 4

%e [5]  0 0 0 3 24

%e [6]  0 0 0 0 33 188

%e [7]  0 0 0 0 13 338 1705

%e [8]  0 0 0 0 0 252 3580 16980

%e [9]  0 0 0 0 0 68 3740 39525 180670

%e [10] 0 0 0 0 0 0 1938 51300 452865 2020120

%e [11] 0 0 0 0 0 0 399 38076 685419 5354832 23478426

%e [12] 0 0 0 0 0 0 0 15180 646415 9095856 65022840 281481880

%e [13] 0 0 0 0 0 0 0 2530 373175 10215450 120872850 807560625 3461873536

%e [14] 0 0 0 0 0 0 0 0 121095 7580040 155282400 1614234960 10224817515 43494961404

%e ...

%o (PARI)

%o T(n,m) = {

%o   if (m <= 1+n\2 || n < 3, return(0));

%o   sum(k=0, m-1, sum(j=0, n-1,

%o      (-1)^((k+j+1)%2) * binomial(k+j,k)*(k+j+1)*(k+j+2)/2*

%o      (binomial(2*n, m-k-1) * binomial(2*m, n-j-1) -

%o       4 * binomial(2*n-1, m-k-2) * binomial(2*m-1, n-j-2))));

%o };

%o concat(vector(14, n, vector(n,m, T(n,m))))  \\ _Gheorghe Coserea_, Jan 08 2017

%Y Right-hand edge is A001506.

%Y See A290326 for a better version.

%K nonn,tabl

%O 1,10

%A _N. J. A. Sloane_, Mar 19 2012

%E a(105)=T(14,14) corrected by _Sean A. Irvine_, Sep 28 2015

%E Name changed by _Gheorghe Coserea_, Jul 23 2017