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%I #29 Sep 29 2018 10:41:34
%S 0,0,0,0,0,1,0,0,0,4,3,0,0,0,3,24,33,13,0,0,0,0,33,188,338,252,68,0,0,
%T 0,0,13,338,1705,3580,3740,1938,399,0,0,0,0,0,252,3580,16980,39525,
%U 51300,38076,15180,2530,0,0,0,0,0,68,3740,39525,180670,452865,685419,646415,373175,121095,16965,0,0,0,0,0,0,1938,51300,452865,2020120,5354832,9095856,10215450,7580040,3585270,981708,118668
%N Triangle read by rows: T(n,k) is the number of c-nets with n+1 faces and k+1 vertices.
%C Row n >= 3 contains 2*n-3 terms.
%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
%H Gheorghe Coserea, <a href="/A290326/b290326.txt">Rows n = 1..103, 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,k) = Sum_{i=0..k-1} Sum_{j=0..n-1} (-1)^(i+j+1) * ((i+j+2)!/(2!*i!*j!)) * (binomial(2*n, k-i-1) * binomial(2*k, n-j-1) - 4 * binomial(2*n-1, k-i-2) * binomial(2*k-1, n-j-2)) for all n >= 3, k >= 3.
%F A106651(n+1) = Sum_{k=1..2*n-3} T(n,k) for n >= 3.
%F A000287(n) = Sum_{i=1+floor((n+2)/3)..floor(2*n/3)-1} T(i,n-i).
%F A001506(n) = T(n,n), A001507(n) = T(n+1,n), A001508(n) = T(n+2,n).
%F A000260(n-2) = T(n, 2*n-3) for n>=3.
%F G.f. y = A(x;t) satisfies: 0 = (t + 1)^3*(x + 1)^3*(t + x + t*x)^3*y^4 + t*(t + 1)^2*x*(x + 1)^2*((4*t^4 + 12*t^3 + 12*t^2 + 4*t)*x^4 + (12*t^4 + 16*t^3 - 4*t^2 - 8*t)*x^3 + (12*t^4 - 4*t^3 - 49*t^2 - 30*t + 3)*x^2 + (4*t^4 - 8*t^3 - 30*t^2 - 21*t)*x + 3*t^2)*y^3 + t^2*(t + 1)*x^2*(x + 1)*((6*t^5 + 18*t^4 + 18*t^3 + 6*t^2)*x^5 + (18*t^5 + 12*t^4 - 30*t^3 - 24*t^2)*x^4 + (18*t^5 - 30*t^4 - 123*t^3 - 58*t^2 + 17*t)*x^3 + (6*t^5 - 24*t^4 - 58*t^3 + 25*t^2 + 56*t)*x^2 + (17*t^3 + 56*t^2 + 48*t + 3)*x + 3*t)*y^2 + t^3*x^3*((4*t^6 + 12*t^5 + 12*t^4 + 4*t^3)*x^6 + (12*t^6 - 36*t^4 - 24*t^3)*x^5 + (12*t^6 - 36*t^5 - 99*t^4 - 26*t^3 + 25*t^2)*x^4 + (4*t^6 - 24*t^5 - 26*t^4 + 81*t^3 + 80*t^2)*x^3 + (25*t^4 + 80*t^3 + 44*t^2 - 14*t)*x^2 + (-14*t^2 - 17*t)*x + 1)*y + t^6*x^6*((t^4 + 2*t^3 + t^2)*x^4 + (2*t^4 - 7*t^3 - 9*t^2)*x^3 + (t^4 - 9*t^3 + 11*t)*x^2 + (11*t^2 + 13*t)*x - 1). - _Gheorghe Coserea_, Sep 29 2018
%e A(x;t) = t^3*x^3 + (4*t^4 + 3*t^5)*x^4 + (3*t^4 + 24*t^5 + 33*t^6 + 13*t^7)*x^5 + ...
%e Triangle starts:
%e n\k [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
%e [1] 0;
%e [2] 0, 0;
%e [3] 0, 0, 1;
%e [4] 0, 0, 0, 4, 3;
%e [5] 0, 0, 0, 3, 24, 33, 13;
%e [6] 0, 0, 0, 0, 33, 188, 338, 252, 68;
%e [7] 0, 0, 0, 0, 13, 338, 1705, 3580, 3740, 1938, 399;
%e [8] 0, 0, 0, 0, 0, 252, 3580, 16980, 39525, 51300, 38076, 15180, 2530;
%e [9] ...
%o (PARI)
%o T(n,k) = {
%o if (n < 3 || k < 3, return(0));
%o sum(i=0, k-1, sum(j=0, n-1,
%o (-1)^((i+j+1)%2) * binomial(i+j, i)*(i+j+1)*(i+j+2)/2*
%o (binomial(2*n, k-i-1) * binomial(2*k, n-j-1) -
%o 4 * binomial(2*n-1, k-i-2) * binomial(2*k-1, n-j-2))));
%o };
%o N=10; concat(concat([0,0,0], apply(n->vector(2*n-3, k, T(n,k)), [3..N])))
%o \\ test 1: N=100; y=x*Ser(vector(N, n, sum(i=1+(n+2)\3, (2*n)\3-1, T(i,n-i)))); 0 == x*(x+1)^2*(x+2)*(4*x-1)*y' + 2*(x^2-11*x+1)*(x+1)^2*y + 10*x^6
%o /*
%o \\ test 2:
%o x='x; t='t; N=44; y=Ser(apply(n->Polrev(vector(2*n-3, k, T(n, k)), 't), [3..N+2]), 'x) * t*x^3;
%o 0 == (t + 1)^3*(x + 1)^3*(t + x + t*x)^3*y^4 + t*(t + 1)^2*x*(x + 1)^2*((4*t^4 + 12*t^3 + 12*t^2 + 4*t)*x^4 + (12*t^4 + 16*t^3 - 4*t^2 - 8*t)*x^3 + (12*t^4 - 4*t^3 - 49*t^2 - 30*t + 3)*x^2 + (4*t^4 - 8*t^3 - 30*t^2 - 21*t)*x + 3*t^2)*y^3 + t^2*(t + 1)*x^2*(x + 1)*((6*t^5 + 18*t^4 + 18*t^3 + 6*t^2)*x^5 + (18*t^5 + 12*t^4 - 30*t^3 - 24*t^2)*x^4 + (18*t^5 - 30*t^4 - 123*t^3 - 58*t^2 + 17*t)*x^3 + (6*t^5 - 24*t^4 - 58*t^3 + 25*t^2 + 56*t)*x^2 + (17*t^3 + 56*t^2 + 48*t + 3)*x + 3*t)*y^2 + t^3*x^3*((4*t^6 + 12*t^5 + 12*t^4 + 4*t^3)*x^6 + (12*t^6 - 36*t^4 - 24*t^3)*x^5 + (12*t^6 - 36*t^5 - 99*t^4 - 26*t^3 + 25*t^2)*x^4 + (4*t^6 - 24*t^5 - 26*t^4 + 81*t^3 + 80*t^2)*x^3 + (25*t^4 + 80*t^3 + 44*t^2 - 14*t)*x^2 + (-14*t^2 - 17*t)*x + 1)*y + t^6*x^6*((t^4 + 2*t^3 + t^2)*x^4 + (2*t^4 - 7*t^3 - 9*t^2)*x^3 + (t^4 - 9*t^3 + 11*t)*x^2 + (11*t^2 + 13*t)*x - 1)
%o */
%Y Cf. A000260, A001506, A001507, A001508.
%Y Rows/Columns sum give A106651 (enumeration of c-nets by the number of vertices).
%Y Antidiagonals sum give A000287 (enumeration of c-nets by the number of edges).
%K nonn,tabf
%O 1,10
%A _Gheorghe Coserea_, Jul 27 2017
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