A001506 -id:A001506 - cp's OEIS frontend

A290326 Triangle read by rows: T(n,k) is the number of c-nets with n+1 faces and k+1 vertices.

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, 0, 0, 13, 338, 1705, 3580, 3740, 1938, 399, 0, 0, 0, 0, 0, 252, 3580, 16980, 39525, 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

Offset: 1

Gheorghe Coserea, Jul 27 2017

Row n >= 3 contains 2*n-3 terms.
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

			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 + ...
Triangle starts:
n\k  [1] [2] [3] [4] [5] [6]  [7]   [8]    [9]    [10]   [11]   [12]   [13]
[1]  0;
[2]  0,  0;
[3]  0,  0,  1;
[4]  0,  0,  0,  4,  3;
[5]  0,  0,  0,  3,  24, 33,  13;
[6]  0,  0,  0,  0,  33, 188, 338,  252,   68;
[7]  0,  0,  0,  0,  13, 338, 1705, 3580,  3740,  1938,  399;
[8]  0,  0,  0,  0,  0,  252, 3580, 16980, 39525, 51300, 38076, 15180, 2530;
[9]  ...

Gheorghe Coserea, Rows n = 1..103, flattened
R. C. Mullin, P. J. Schellenberg, The enumeration of c-nets via quadrangulations, J. Combinatorial Theory 4 1968 259--276. MR0218275 (36 #1362).

Cf. A000260, A001506, A001507, A001508.
Rows/Columns sum give A106651 (enumeration of c-nets by the number of vertices).
Antidiagonals sum give A000287 (enumeration of c-nets by the number of edges).

T(n,k) = {
  if (n < 3 || k < 3, return(0));
  sum(i=0, k-1, sum(j=0, n-1,
     (-1)^((i+j+1)%2) * binomial(i+j, i)*(i+j+1)*(i+j+2)/2*
     (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))));
};
N=10; concat(concat([0,0,0], apply(n->vector(2*n-3, k, T(n,k)), [3..N])))
\\ 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
/*
\\ test 2:
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;
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)
*/

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.
A106651(n+1) = Sum_{k=1..2*n-3} T(n,k) for n >= 3.
A000287(n) = Sum_{i=1+floor((n+2)/3)..floor(2*n/3)-1} T(i,n-i).
A001506(n) = T(n,n), A001507(n) = T(n+1,n), A001508(n) = T(n+2,n).
A000260(n-2) = T(n, 2*n-3) for n>=3.
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

A106651 c(n) = number of c-nets on n vertices.

Original entry on oeis.org

1, 1, 7, 73, 879, 11713, 167423, 2519937, 39458047, 637446145, 10561615871, 178683815937, 3076487458815, 53766284722177, 951817354412031, 17039752595865601, 308068940431556607, 5618467344224354305

Offset: 3

Daniel Johannsen (johannse(AT)informatik.hu-berlin.de), May 12 2005

Definition of c-net: a 3-connected planar map, rooted by a directed edge on the outer face.

			c(0)=c(1)=1 because the only c-nets on 3 respectively 4 vertices are the complete graphs.

Gheorghe Coserea, Table of n, a(n) for n = 3..302
M. Bodirsky, C. Groepl, D. Johannsen and M. Kang, A Direct Decomposition of 3-connected Planar Graphs, conference paper (FPSAC05).
Gheorghe Coserea, Algebraic equation for g.f.
R. C. Mullin, P. J. Schellenberg, The enumeration of c-nets via quadrangulations, J. Combinatorial Theory 4 1968 259--276. MR0218275 (36 #1362).

Cf. A001506, A001507, A001508, A285165, A290326.

c[0] = 1; c[1] = 1; c[2] = 7; c[3] = 73; c[4] = 879; c[5] = 11713; c[6] = 167423; c[7] = 2519937; c[n_] := c[n] = ( (-189665280 + 134270976 n - 31309824 n^2 + 2408448 n^3) c[n - 7] + (-479162880 + 376680448 n - 98932224 n^2 + 8692736 n^3) c[n - 6] + (-446660160 + 384601888 n - 112131264 n^2 + 11026784 n^3) c[n - 5] + (-183645792 + 168826836 n - 52598160 n^2 + 5361276 n^3) c[n - 4] + (-25324080 + 24563948 n - 6853668 n^2 + 418816 n^3) c[n - 3] + (1156086 - 2064937 n + 1206966 n^2 - 180467 n^3) c[n - 2] + (-3192 + 4842 n - 29796 n^2 + 18930 n^3) c[n - 1] ) / (126 + 693 n + 1134 n^2 + 567 n^3);

x='x; y='y;
system("wget http://oeis.org/A106651/a106651.txt");
Fxy = read("a106651.txt");
seq(N) = {
  my(y0 = 1 + O('x^N), y1=0);
  for (k = 1, N,
    y1 = y0 - subst(Fxy, y, y0)/subst(deriv(Fxy, y), y, y0);
    if (y1 == y0, break()); y0 = y1);
  Vec(y0);
};
seq(18)  \\ Gheorghe Coserea, Jan 08 2017

A290326(n,k) = {
  if (n < 3 || k < 3, return(0));
  sum(i=0, k-1, sum(j=0, n-1,
     (-1)^((i+j+1)%2) * binomial(i+j, i)*(i+j+1)*(i+j+2)/2*
     (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))));
};
a(n) = if (n==3, 1, sum(k = (n+3)\2, 2*n-5, A290326(n-1, k)));
vector(18, n, a(n+2)) \\ Gheorghe Coserea, Jul 28 2017

c(0)=1, c(1) = 1, c(2) = 7, c(3) = 73, c(4) = 879, c(5) = 11713, c(6) = 167423, c(7) = 2519937, c(n) = ( (-189665280 + 134270976 n - 31309824 n^2 + 2408448 n^3) c(n-7) + (-479162880 + 376680448 n - 98932224 n^2 + 8692736 n^3) c(n-6) + (-446660160 + 384601888 n - 112131264 n^2 + 11026784 n^3) c(n-5) + (-183645792 + 168826836 n - 52598160 n^2 + 5361276 n^3) c(n-4) + (-25324080 + 24563948 n - 6853668 n^2 + 418816 n^3) c(n-3) + (1156086 - 2064937 n + 1206966 n^2 - 180467 n^3) c(n-2) + (-3192 + 4842 n - 29796 n^2 + 18930 n^3) c(n-1) ) / (126 + 693 n + 1134 n^2 + 567 n^3). Generating function C(t)=sum_(n>=0){c(n-3)t^n} implicitly given by: 0 = -1 + C(t) + 36 t - 43 C(t) t + 6 C(t)^2 t + 131 t^2 - 337 C(t) t^2 + 218 C(t)^2 t^2 + 12 C(t)^3 t^2 + 350 t^3 - 1021 C(t) t^3 + 894 C(t)^2 t^3 - 228 C(t)^3 t^3 + 8 C(t)^4 t^3 + 540 t^4 - 1828 C(t) t^4 + 2190 C(t)^2 t^4 - 988 C(t)^3 t^4 + 72 C(t)^4 t^4 + 616 t^5 - 2404 C(t) t^5 + 3284 C(t)^2 t^5 - 1756 C(t)^3 t^5 + 264 C(t)^4 t^ 5 + 536 t^6 - 2128 C(t) t^6 + 3120 C(t)^2 t^6 - 2032 C(t)^3 t^6 + 504 C(t)^4 t^6 + 304 t^7 - 1344 C(t) t^7 + 2304 C(t)^2 t^7 - 1792 C(t)^3 t^7 + 528 C(t)^4 t^7 + 160 t^8 - 768 C(t) t^8 + 1344 C(t)^2 t^8 - 1024 C(t)^3 t^8 + 288 C(t)^4 t^8 + 64 t^9 - 256 C(t) t^9 + 384 C(t)^2 t^9 - 256 C(t)^3 t^9 + 64 C(t)^4 t^9. Explicit generating function can be obtained using Mathematica.

Mathematica code improved by David Radcliffe, Feb 12 2011

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.

Original entry on oeis.org

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

N. J. A. Sloane, Mar 19 2012

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

			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
...

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).

Right-hand edge is A001506.

See A290326 for a better version.

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

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

a(105)=T(14,14) corrected by Sean A. Irvine, Sep 28 2015

Name changed by Gheorghe Coserea, Jul 23 2017

cp's OEIS Frontend

A290326 Triangle read by rows: T(n,k) is the number of c-nets with n+1 faces and k+1 vertices.

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A106651 c(n) = number of c-nets on n vertices.

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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.

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