A280786 -id:A280786 - cp's OEIS frontend

A261070 Irregular triangle read by rows: T(n,k) is the number of arrangements of n circles with 2k intersections (using the same rules as A250001).

1, 1, 2, 1, 4, 4, 2, 4, 9, 15, 15, 31, 24, 35, 44, 20, 50

Offset: 0

Benoit Jubin, Aug 08 2015

Length of n-th row: 1 + (n-1)n/2 (for a configuration for T(n,(n-1)n/2), consider n circles of radius 1 and centers at (k/n,0) for 1<=k<=n).

The generating function down the column k=1 is 1+z^2 *C^2(z) *[C^2(z)+C(z^2)]/ (2*[1-z*C(z)]) = 1+ z^2 +4*z^3 +15*z^4+ 50*z^5+...where C(z) = 1+z+2*z^2+4*z^3+... is the g.f. of A000081 divided by z; eq. (78) in arXiv:1603.00077. - R. J. Mathar, Mar 05 2016

			n\k 0  1  2  3  4  5  6
0   1
1   1
2   2  1
3   4  4  2  4
4   9 15 15 31 24 35 44
5  20 50  .  .  .  .  .  .  .  .  .

R. J. Mathar, Topologically Distinct Sets of Non-intersecting Circles in the Plane, arXiv:1603.00077 [math.CO], 2016.

Row sums give A250001.

Cf. A000081, A152947, A249752, A252158, A280786 (column k=1)

A250001(n) = Sum_{k>=0} T(n,k).

A000081(n+1) = T(n,0).

T(4,2)..T(5,0) (6 terms) from Travis Vasquez, Nov 28 2024

A280788 Convolution of A000081 and A027852, shifted by 3 leading zeros.

Original entry on oeis.org

1, 2, 6, 15, 41, 106, 284, 750, 2010, 5382, 14523, 39290, 106854, 291552, 798675, 2194828, 6051153, 16730373, 46383002, 128910484, 359115067, 1002575810, 2804667061, 7860780578, 22070885735, 62071872704, 174842835886, 493217417610

Offset: 0

N. J. A. Sloane, Jan 20 2017

nonn

R. J. Mathar, Topologically Distinct Sets of Non-intersecting Circles in the Plane, arXiv:1603.00077 [math.CO], 2016. See (71).

Cf. A269800, A280786, A280787, A027852.

A280788 := proc(N::integer)
    if N = 0 then
        1;
    else
        add(A000081(Nprime+1)*A027852(N-Nprime+2),Nprime=0..N) ;
    end if;
end proc:
seq(A280788(n),n=0..30) ; # R. J. Mathar, Mar 06 2017

a81[n_] := a81[n] = If[n <= 1, n, Sum[a81[n - j]*DivisorSum[j, #*a81[#]&], {j, n - 1}]/(n - 1)];
A027852[n_] := Module[{dh = 0, np}, For[np = 0, np <= n, np++, dh = a81[np] * a81[n - np] + dh]; If[EvenQ[n], dh = a81[n/2] + dh]; dh/2];
A280788[n_] := If[n == 0, 1, Sum[a81[np+1]*A027852[n-np+2], {np, 0, n}]];
Table[A280788[n], {n, 0, 27}] (* Jean-François Alcover, Nov 23 2017, from Maple *)

A280787 Triangle read by rows: number of topologically distinct sets of n circles with one pair intersecting, by number of factors.

Original entry on oeis.org

1, 3, 1, 10, 4, 1, 30, 15, 4, 1, 91, 50, 16, 4, 1, 268, 162, 55, 16, 4, 1, 790, 506, 185, 56, 16, 4, 1, 2308, 1558, 594, 190, 56, 16, 4, 1, 6737, 4727, 1878, 617, 191, 56, 16, 4, 1, 19609, 14227, 5825, 1970, 622, 191, 56, 16, 4, 1

Offset: 2

N. J. A. Sloane, Jan 20 2017

			Triangle begins:
     1;
     3,    1;
    10,    4,   1;
    30,   15,   4,   1;
    91,   50,  16,   4,  1;
   268,  162,  55,  16,  4,  1;
   790,  506, 185,  56, 16,  4, 1;
  2308, 1558, 594, 190, 56, 16, 4, 1;
...

R. J. Mathar, Topologically Distinct Sets of Non-intersecting Circles in the Plane, arXiv:1603.00077 [math.CO], 2016.

Row sums give A280786.

a81[n_] := a81[n] = If[n <= 1, n, Sum[a81[n - j]*DivisorSum[j, #1*a81[#1] &], {j, n - 1}]/(n - 1)];
A027852[n_] := Module[{dh = 0, np}, For[np = 0, np <= n, np++, dh = a81[np]*a81[n - np] + dh]; If[EvenQ[n], dh = a81[n/2] + dh]; dh/2];
A280788[n_] := If[n == 0, 1, Sum[a81[np+1]*A027852[n-np+2], {np, 0, n}]];
t[n_] := t[n] = Module[{d, j}, If[n == 1, 1, Sum[Sum[d*t[d], {d, Divisors[j]}]*t[n - j], {j, 1, n - 1}]/(n - 1)]];
b[1, 1, 1] = 1;
b[n_, i_, p_] := b[n, i, p] = If[p > n, 0, If[n == 0, 1, If[Min[i, p] < 1, 0, Sum[b[n - i*j, i - 1, p - j]*Binomial[t[i] + j - 1, j], {j, 0, Min[n/i, p]}]]]]; A033185[n_, k_] := b[n, n, k];
A280786[n_] := If[n < 2, 0, Sum[A280787[n, f], {f, 1, n - 1}]];
A280787[n_, f_] := A280787[n, f] = Module[{ct}, Which[f == n, Return[0], f == n - 1, Return[1], f == 1, Return[A280786[n - 1] + A280788[n - 2]], True, ct = 0; Do[ct += A280787[np, 1]*A033185[n - np, f - 1], {np, 1, n - 1}]]; ct];
Table[A280787[n, f], {n, 2, 11}, {f, 1, n - 1}] // Flatten (* Jean-François Alcover, Nov 23 2017, after R. J. Mathar and Alois P. Heinz *)

cp's OEIS Frontend

A261070 Irregular triangle read by rows: T(n,k) is the number of arrangements of n circles with 2k intersections (using the same rules as A250001).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A280788 Convolution of A000081 and A027852, shifted by 3 leading zeros.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

A280787 Triangle read by rows: number of topologically distinct sets of n circles with one pair intersecting, by number of factors.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica