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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.

%I A210252 #46 Feb 18 2025 05:56:33
%S A210252 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 A210252 0,0,0,0,252,3580,16980,0,0,0,0,0,68,3740,39525,180670,0,0,0,0,0,0,
%U A210252 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 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.
%C A210252 c-nets are 3-connected rooted planar maps. This array also counts simple triangulations.
%C A210252 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 A210252 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 A210252 Gheorghe Coserea, <a href="/A210252/b210252.txt">Rows n = 1..100, flattened</a>
%H A210252 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 A210252 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 A210252 Triangle begins:
%e A210252 n\k
%e A210252 [1]  0
%e A210252 [2]  0 0
%e A210252 [3]  0 0 1
%e A210252 [4]  0 0 0 4
%e A210252 [5]  0 0 0 3 24
%e A210252 [6]  0 0 0 0 33 188
%e A210252 [7]  0 0 0 0 13 338 1705
%e A210252 [8]  0 0 0 0 0 252 3580 16980
%e A210252 [9]  0 0 0 0 0 68 3740 39525 180670
%e A210252 [10] 0 0 0 0 0 0 1938 51300 452865 2020120
%e A210252 [11] 0 0 0 0 0 0 399 38076 685419 5354832 23478426
%e A210252 [12] 0 0 0 0 0 0 0 15180 646415 9095856 65022840 281481880
%e A210252 [13] 0 0 0 0 0 0 0 2530 373175 10215450 120872850 807560625 3461873536
%e A210252 [14] 0 0 0 0 0 0 0 0 121095 7580040 155282400 1614234960 10224817515 43494961404
%e A210252 ...
%o A210252 (PARI)
%o A210252 T(n,m) = {
%o A210252   if (m <= 1+n\2 || n < 3, return(0));
%o A210252   sum(k=0, m-1, sum(j=0, n-1,
%o A210252      (-1)^((k+j+1)%2) * binomial(k+j,k)*(k+j+1)*(k+j+2)/2*
%o A210252      (binomial(2*n, m-k-1) * binomial(2*m, n-j-1) -
%o A210252       4 * binomial(2*n-1, m-k-2) * binomial(2*m-1, n-j-2))));
%o A210252 };
%o A210252 concat(vector(14, n, vector(n,m, T(n,m))))  \\ _Gheorghe Coserea_, Jan 08 2017
%Y A210252 Right-hand edge is A001506.
%Y A210252 See A290326 for a better version.
%K A210252 nonn,tabl
%O A210252 1,10
%A A210252 _N. J. A. Sloane_, Mar 19 2012
%E A210252 a(105)=T(14,14) corrected by _Sean A. Irvine_, Sep 28 2015
%E A210252 Name changed by _Gheorghe Coserea_, Jul 23 2017