A000287 -id:A000287 - 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

A343871 Number of labeled 3-connected planar graphs with n edges.

Original entry on oeis.org

1, 0, 15, 70, 432, 4320, 30855, 294840, 2883240, 28175952, 310690800, 3458941920, 40459730640, 499638948480, 6324655705200, 83653192972800, 1145266802114400, 16145338385736000, 235579813593453000, 3535776409508703360, 54571687068401395200, 866268656574795936000

Offset: 6

Andrew Howroyd, May 05 2021

nonn

Andrew Howroyd, Table of n, a(n) for n = 6..200
M. Bodirsky, C. Groepl and M. Kang, Generating Labeled Planar Graphs Uniformly At Random, Theoretical Computer Science, Volume 379, Issue 3, 15 June 2007, pp. 377-386.

Cf. A000287, A002840 (unlabeled case), A096330, A290326, A291841, A338414.

Q(n,k) = { \\ c-nets with n-edges, k-vertices (see A290326)
  if (k < 2+(n+2)\3 || k > 2*n\3, return(0));
  sum(i=2, k, sum(j=k, n, (-1)^((i+j+1-k)%2)*binomial(i+j-k,i)*i*(i-1)/2*
  (binomial(2*n-2*k+2,k-i)*binomial(2*k-2, n-j) -
  4*binomial(2*n-2*k+1, k-i-1)*binomial(2*k-3, n-j-1))));
};
a(n)={sum(k=2+(n+2)\3, 2*n\3, k!*Q(n,k))/(4*n)} \\ Andrew Howroyd, May 05 2021

a(n) = Sum_{k=2+floor((n+2)/3)..floor(2*n/3)} k!*A290326(n-k+1, k-1)/(4*n).

A047051 Prime alternating tangle types (of knots) with n crossings.

Original entry on oeis.org

1, 2, 4, 10, 29, 98, 372, 1538, 6755, 30996, 146982, 715120, 3552254, 17951322, 92045058, 477882876, 2508122859, 13289437362, 71010166670, 382291606570, 2072025828101, 11298920776704, 61954857579594, 341427364138880

Offset: 1

Valery A. Liskovets

G.f. is related to the classes of 2- and 3-connected planar maps with n edges. Further terms are known.

C. Sundberg and M. Thistlethwaite, The rate of growth of the number of prime alternating links and tangles, Pacif. J. Math., 182, No 2 (1998), 329-358.

Herman Jamke (hermanjamke(AT)fastmail.fm), May 05 2007, Table of n, a(n) for n = 1..50
S. R. Finch, Knots, links and tangles
S. R. Finch, Knots, links and tangles, Aug 08 2003. [Cached copy, with permission of the author]
C. Sundberg and M. Thistlethwaite, The rate of growth of the number of prime alternating links and tangles, Pacif. J. Math., 182, No 2 (1998), 329-358.
P. Zinn-Justin and J.-B. Zuber, Matrix integrals and the generation and counting of virtual tangles and links, p. 12.
P. Zinn-Justin, The General O(n) Quartic Matrix Model and its application to Counting Tangles and Links

Cf. A002864, A000139, A000287.

max = 24; Clear[a, eq, s]; gf = Sum[a[k]*x^k, {k, 0, max}]; a[0] = 0; a[1] = 1; a[2] = 2; coes = CoefficientList[(x^4 - 2*x^3 + x^2)*gf^5 + (8*x^4 - 14*x^3 + 8*x^2 - 2*x)*gf^4 + (25*x^4 - 16*x^3 - 14*x^2 + 8*x + 1)*gf^3 + (38*x^4 + 15*x^3 - 30*x^2 - x + 2)*gf^2 + (28*x^4 + 36*x^3 - 5*x^2 - 12*x + 1)*gf + 8*x^4 + 17*x^3 + 8*x^2 - x, x]; eq[n_] := eq[n] = If[n == 1, Thread[Drop[coes, 3] == 0], eq[n-1] /. s[n-1] // First]; s[n_] := s[n] = (Print["n = ", n]; Solve[eq[n][[n]], a[n+2]]); sol = Table[s[n], {n, 1, max-2}] // Flatten; Table[a[n], {n, 1, max}] /. sol (* Jean-François Alcover, Apr 15 2014 *)

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), May 05 2007

A111157 Numbers k such that sequence A_k in this database does not contain a prime.

Original entry on oeis.org

4, 7, 12, 35, 56, 66, 82, 118, 132, 141, 143, 144, 145, 152, 156, 173, 252

Offset: 1

N. J. A. Sloane, following a suggestion of Alonso del Arte, Oct 21 2005

That is, the complete sequence A_k, not just the terms that are shown in the entry, does not contain a prime or the negative of a prime. The only exception to this is for sequences with keyword "dead", when only the visible terms are to be considered.
No other terms below 287. Is 287 a member? This requires proof.
I've checked A000287 up to a(25000) and can report that I found no prime. - Robert G. Wilson v, Jun 23 2014
A search of every term of A000287 of form a(2^k-2) has yielded no primes up to a(2^24-2). It shouldn't have any odd terms outside this subsequence, and assuming that its terms' factors are otherwise random, it has a 2.15*10^-8 probability of containing any prime at all. After A000287, the next difficult sequence is A000373, which should contain infinitely many primes, but has none within its first 10000 terms. - Matthew House, Nov 02 2024

PrimeFan, Esoteric Integer Sequences (Gives a different version.)
PrimeFan, Esoteric Integer Sequences [Cached copy]
Index entries for sequences whose definition involves A_n (or An).

Cf. A111198.

a(16) = 173 inserted by Matthew House, Nov 02 2024

A210685 a(1)=-1, a(2)=2, thereafter a(n) = (1/(2n))*((7n-22)a(n-1)+2(2n-1)a(n-2)).

Original entry on oeis.org

-1, 2, -2, 2, -1, 2, 2, 8, 22, 68, 212, 678, 2207, 7298, 24458, 82928, 284066, 981884, 3421316, 12007556, 42416486, 150718772, 538421588, 1932856592, 6969847484, 25237057112, 91729488352, 334589415278, 1224445617887, 4494622119426, 16545704494682, 61071167108960, 225982053183050

Offset: 1

N. J. A. Sloane, Mar 28 2012

sign

Liu Yanpei, On the number of rooted c-nets, J. Combin. Theory, B 36 (1984), 118-123. See the sequence R_n.

Cf. A000287.

R:=proc(n) option remember; if n=1 then -1 elif n=2 then 2 else
(1/(2*n))*((7*n-22)*R(n-1)+2*(2*n-1)*R(n-2)); fi; end;
[seq(R(n),n=1..40)];

a[n_] := a[n] = Which[n == 1, -1, n == 2, 2, True, (1/(2*n))*((7*n-22)*a[n-1]+2*(2*n-1)*a[n-2])]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Mar 06 2014, after Maple *)

A000287(n) = (-1)^n*2 + a(n-1).

cp's OEIS Frontend

A241602 Erroneous version of A000287.

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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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A343871 Number of labeled 3-connected planar graphs with n edges.

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A047051 Prime alternating tangle types (of knots) with n crossings.

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A111157 Numbers k such that sequence A_k in this database does not contain a prime.

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A210685 a(1)=-1, a(2)=2, thereafter a(n) = (1/(2n))*((7n-22)a(n-1)+2(2n-1)a(n-2)).

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