A050534 -id:A050534 - cp's OEIS frontend

A144161 Triangle read by rows: T(n,k) = number of simple graphs on n labeled nodes with k edges that are node-disjoint unions of undirected cycle subgraphs.

1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 4, 3, 1, 0, 0, 10, 15, 12, 1, 0, 0, 20, 45, 72, 70, 1, 0, 0, 35, 105, 252, 490, 465, 1, 0, 0, 56, 210, 672, 1960, 3720, 3507, 1, 0, 0, 84, 378, 1512, 5880, 16740, 31563, 30016, 1, 0, 0, 120, 630, 3024, 14700, 55800, 157815, 300160, 286884

Offset: 0

Alois P. Heinz, Sep 12 2008

			T(4,3) = 4, because there are 4 simple graphs with 3 edges that are node-disjoint unions of undirected cycle subgraphs:
  .1.2. .1.2. .1-2. .1-2.
  ../|. .|\.. ..\|. .|/..
  .3-4. .3-4. .3.4. .3.4.
T(6,6) = C(6,3)/2+5!/2 = 70.
Triangle begins:
  1;
  1, 0;
  1, 0, 0;
  1, 0, 0,  1;
  1, 0, 0,  4,  3;
  1, 0, 0, 10, 15, 12;
  1, 0, 0, 20, 45, 72, 70;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0, 1+2, 3-4 give: A000012, A000004, A000292, A050534.
Main diagonal gives A001205.
Row sums give: A108246.
Cf. A007318, A000142.

T:= proc(n,k) option remember; local i,j; if k=0 then 1 elif k<0 or n

T[n_, k_] := T[n, k] = Module[{i, j}, If[k == 0, 1, If[k < 0 || n < k, 0, T[n - 1, k] + Sum[Product[n - i, {i, 1, j}]*T[n - 1 - j, k - j - 1], {j, 2, k}]/2 ]]]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 27 2013, translated from Maple *)

from sympy.core.cache import cacheit
from operator import mul
from functools import reduce
@cacheit
def T(n, k): return 1 if k==0 else 0 if k<0 or nIndranil Ghosh, Aug 07 2017

T(n,0) = 1, T(n,k) = 0 if k<0 or n

A144209 Triangle T(n,k), n>=0, 0<=k<=n, read by rows: T(n,k) = number of simple graphs on n labeled nodes with k edges where each maximally connected subgraph consists of a single node or has a unique cycle of length 4.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 3, 1, 0, 0, 0, 15, 60, 1, 0, 0, 0, 45, 360, 1080, 1, 0, 0, 0, 105, 1260, 7560, 20580, 1, 0, 0, 0, 210, 3360, 30240, 164640, 430080, 1, 0, 0, 0, 378, 7560, 90720, 740880, 3873240, 9920232, 1, 0, 0, 0, 630, 15120, 226800, 2469600, 19367460, 99406440, 252000000

Offset: 0

Alois P. Heinz, Sep 14 2008

			T(5,4) = 15 = 5*3, because there are 5 possibilities for a single node and T(4,4) = 3:
.1-2. .1-2. .1.2.
.|.|. ..X.. .|X|.
.3-4. .3-4. .3.4.
Triangle begins:
1;
1, 0;
1, 0, 0;
1, 0, 0, 0;
1, 0, 0, 0,  3;
1, 0, 0, 0, 15, 60;

Alois P. Heinz, Rows n = 0..140, flattened

Columns 0, 1+2+3, 4 give: A000012, A000004, A050534.
Main diagonal gives A065889.
Row sums give A144210.
Cf. A007318.

T:= proc(n,k) option remember; if k=0 then 1 elif k<0 or n

T[n_, k_] := T[n, k] = Which[k == 0, 1, k < 0 || n < k, 0, k == n, 3*Binomial[n-1, 3]*n^(n-4), True, T[n-1, k] + Sum[Binomial[n-1, j]*T[j+1, j+1]*T[n-1-j, k-j-1], {j, 3, k-1}]]; Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Aug 29 2014, translated from Maple *)

T(n,0) = 1, T(n,k) = 0 if k<0 or n

A198148 a(n) = n(n+2)(9 - 7*(-1)^n)/16.

Original entry on oeis.org

0, 3, 1, 15, 3, 35, 6, 63, 10, 99, 15, 143, 21, 195, 28, 255, 36, 323, 45, 399, 55, 483, 66, 575, 78, 675, 91, 783, 105, 899, 120, 1023, 136, 1155, 153, 1295, 171, 1443, 190, 1599, 210, 1763, 231, 1935, 253, 2115, 276, 2303, 300, 2499, 325

Offset: 0

Paul Curtz, Oct 21 2011

See, in A181318(n), A060819(n)*A060819(n+p): A060819(n)^2, A064038(n), a(n), A160050(n), A061037(n), A178242(n). The second differences a(n+2)-2*a(n+1)+a(n) = -5, 16, -26, 44, -61, 86, -110, 142, -173, 212, -250, 296, -341, 394, -446, 506, taken modulo 9 are periodic with the palindromic period 4, 7, 1, 8, 2, 5, 7, 7, 7, 5, 2, 8, 1, 7, 4.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Cf. quadrisections A014105, A001539, A000384, A141759.

Cf. A060819, A000217, A000466, A142705, A000034, A005563, A010689, A028552, A050534.

List([0..60], n -> n*(n+2)*(9-7*(-1)^n)/16); # G. C. Greubel, Feb 21 2019

[n*(n+2)*(9-7*(-1)^n)/16: n in [0..60]]; // Vincenzo Librandi, Nov 25 2011

A198148:=n->n*(n+2)*(9-7*(-1)^n)/16; seq(A198148(k), k=0..100); # Wesley Ivan Hurt, Oct 16 2013

LinearRecurrence[{0,3,0,-3,0,1},{0,3,1,15,3,35},60] (* Vincenzo Librandi, Nov 25 2011 *)

a(n)=n*(n+2)*(9-7*(-1)^n)/16 \\ Charles R Greathouse IV, Oct 16 2015

[n*(n+2)*(9-7*(-1)^n)/16 for n in (0..60)] # G. C. Greubel, Feb 21 2019

a(n) = A060819(n)*A060819(n+2).
a(2n) = n*(n+1)/2 = A000217(n).
a(2n+1) = (2*n+1)*(2*n+3) = A000466(n+1).
a(n) = 3*a(n-2) - 3*a(n-4) + a(n-6), n>5.
a(n+1) - a(n) = (7*(-1)^n *(2*n^2+6*n+3) +18*n +27)/16.
a(n) = A142705(n) / A000034(n+1).
a(n) = A005563(n) / A010689(n+1). - Franklin T. Adams-Watters, Oct 21 2011
G.f. x*(3 +x +6*x^2 -x^4)/(1-x^2)^3. - R. J. Mathar, Oct 25 2011
a(n)*a(n+1) = a(A028552(n)) = A050534(n+2). - Bruno Berselli, Oct 26 2011
a(n) = numerator( binomial((n+2)/2,2) ). - Wesley Ivan Hurt, Oct 16 2013
E.g.f.: x*((24+x)*cosh(x) + (3+8*x)*sinh(x))/8. - G. C. Greubel, Sep 20 2018
Sum_{n>=1} 1/a(n) = 5/2. - Amiram Eldar, Aug 12 2022

A234251 Triangle T(n, k) = Number of ways to choose k points from an n X n X n triangular grid so that no three of them form a 2 X 2 X 2 subtriangle. Triangle T read by rows.

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 6, 15, 16, 6, 1, 10, 45, 111, 156, 120, 42, 2, 1, 15, 105, 439, 1191, 2154, 2583, 1977, 885, 189, 9, 1, 21, 210, 1305, 5565, 17052, 38337, 63576, 77208, 67285, 40512, 15750, 3480, 333, 9, 1, 28, 378, 3240, 19620, 88590, 307362, 833228, 1779219

Offset: 1

Heinrich Ludwig, Feb 06 2014

n starts from 1. The maximal number of points that can be chosen from a grid of side n, so that no three of them are forming a subtriangle of side 2, is A007980(n - 1). So k ranges from 0 to A007980(n - 1).
Column #2 (k = 1) is A000217.
Column #3 (k = 2) is A050534.
Column #4 (k = 3) is A234250.

			Triangle begins
  1,  1;
  1,  3,   3;
  1,  6,  15,  16,    6;
  1, 10,  45, 111,  156,  120,   42,    2;
  1, 15, 105, 439, 1191, 2154, 2583, 1977, 885, 189, 9;
  ...
There are no more than T(4, 7) = 2 ways to choose 7 points (X) from a 4 X 4 X 4 grid so that no 3 of them form a 2 X 2 X 2 subtriangle:
        X              X
       X .            . X
      . X X          X X .
     X X . X        X . X X

Heinrich Ludwig, Table of n, a(n) for n = 1..133

Cf. A000217, A050534, A234250.

A243214 Number of ways to place 5 points on a triangular grid of side n, so that no three of them are vertices of an equilateral triangle with sides parallel to the grid.

Original entry on oeis.org

0, 63, 1566, 13971, 77124, 319206, 1083723, 3181401, 8344854, 20006349, 44548227, 93248628, 185176866, 351410664, 640972980, 1129067352, 1928196432, 3203016813

Offset: 3

Heinrich Ludwig, Jun 10 2014

Cf. A243211, A243210, A000217, A050534, A243212, A243213.

a(n) = (n^10 + 5*n^9 - 10*n^8)/3840 + O(n^7)

A053527 Number of bipartite graphs with 4 edges on nodes {1..n}.

Original entry on oeis.org

0, 0, 0, 0, 3, 140, 1125, 5355, 19075, 56133, 143955, 332475, 706860, 1404975, 2640638, 4733820, 8149050, 13543390, 21825450, 34227018, 52388985, 78463350, 115233195, 166252625, 236008773, 330108075, 455489125, 620664525, 835994250

Offset: 0

N. J. A. Sloane, Jan 16 2000

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.5.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Column k=4 of A117279.

Cf. A000217 (1 edge), A050534 (2 edges), A053526 (3 edges).

List([0..30], n-> Binomial(n,4)*(n+2)*(n^3-5*n-36)/16 ) # G. C. Greubel, May 15 2019

[(n^5-4*n^4-n^3+16*n^2-12*n)*(n^3-5*n-36)/384: n in [0..30]]; // Vincenzo Librandi, May 08 2012

CoefficientList[Series[x^4*(3+113*x-27*x^2+18*x^3-2*x^4)/(1-x)^9, {x,0, 30}], x] (* Vincenzo Librandi, May 08 2012 *)

{a(n) = binomial(n,4)*(n+2)*(n^3-5*n-36)/16}; \\ G. C. Greubel, May 15 2019

[binomial(n,4)*(n+2)*(n^3-5*n-36)/16 for n in (0..30)] # G. C. Greubel, May 15 2019

a(n) = (n-3)*(n-2)*(n-1)*n*(n+2)*(n^3-5*n-36)/384.
G.f.: x^4*(3+113*x-27*x^2+18*x^3-2*x^4)/(1-x)^9. - Colin Barker, May 08 2012
E.g.f.: x^4*(48 + 400*x + 176*x^2 + 24*x^3 + x^4)*exp(x)/384. - G. C. Greubel, May 15 2019

A144163 Triangle T(n,k), n>=0, 0<=k<=n, read by rows: T(n,k) = number of simple graphs on n labeled nodes with k edges where each maximally connected subgraph is either a tree or a cycle.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 3, 3, 1, 1, 6, 15, 20, 3, 1, 10, 45, 120, 150, 12, 1, 15, 105, 455, 1185, 1473, 70, 1, 21, 210, 1330, 5565, 14469, 18424, 465, 1, 28, 378, 3276, 19635, 81060, 213990, 280200, 3507, 1, 36, 630, 7140, 57393, 334656, 1385076, 3732300, 5029218, 30016

Offset: 0

Alois P. Heinz, Sep 12 2008

			T(4,3) = 20, because there are 20 simple graphs on 4 labeled nodes with 3 edges, where each maximally connected subgraph is either a tree or a cycle, 16 of these graphs consist of a single tree with 4 nodes and 4 consist of a cycle with 3 and a tree with 1 node:
  .1-2. .1-2. .1.2. .1.2. .1-2. .1-2. .1-2. .1-2. .1-2. .1.2.
  .|\.. ../|. ..\|. .|/.. .|... ...|. ../.. ..\.. .|.|. .|.|.
  .4.3. .4.3. .4-3. .4-3. .4-3. .4-3. .4-3. .4-3. .4.3. .4-3.
  .
  .1.2. .1.2. .1-2. .1.2. .1.2. .1.2. .1.2. .1.2. .1-2. .1-2.
  .|/|. .|\|. ..X.. ..X|. ..X.. .|X.. ../|. .|\.. .|/.. ..\|.
  .4.3. .4.3. .4.3. .4.3. .4-3. .4.3. .4-3. .4-3. .4.3. .4.3.
Triangle begins:
  1;
  1,  0;
  1,  1,  0;
  1,  3,  3,   1;
  1,  6, 15,  20,   3;
  1, 10, 45, 120, 150, 12;

Alois P. Heinz, Rows n = 0..140, flattened
Index entries for sequences related to trees

Columns k=0-3 give: A000012, A000217, A050534, A093566.
Main diagonal gives A001205.
Row sums give A144164.
Cf. A138464, A144161, A007318, A000142.

f:= proc(n,k) option remember; local j; if k=0 then 1 elif k<0 or n<=k then 0 elif k=n-1 then n^(n-2) else add(binomial(n-1,j) *f(j+1,j) *f(n-1-j,k-j), j=0..k) fi end:
c:= proc(n,k) option remember; local i,j; if k=0 then 1 elif k<0 or n

f[n_, k_] := f[n, k] = Which[k == 0, 1, k<0 || n <= k, 0, k == n-1, n^(n-2), True, Sum[Binomial[n-1, j]*f[j+1, j]*f[n-1-j, k-j], {j, 0, k}]]; c[n_, k_] := c[n, k] = Which[k == 0, 1 , k<0 || nJean-François Alcover, Jan 21 2014, translated from Alois P. Heinz's Maple code *)

T(n,k) = A138464(n,k) + Sum_{j=3..k} C(n,j) A138464(n-j,k-j) A144161 (j,j).

Previous Showing 31-40 of 54 results. Next

cp's OEIS Frontend

A243211 Triangle T(n, k) = Numbers of ways to place k points on a triangular grid of side n so that no three of them are vertices of an equilateral triangle with sides parallel to the grid. Triangle read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A243212 Number of ways to place 3 points on a triangular grid of side n so that no three of them are vertices of an equilateral triangle with sides parallel to the grid.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A243213 Number of ways to place 4 points on a triangular grid of side length n so that no three of them are vertices of an equilateral triangle with sides parallel to the grid.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A144161 Triangle read by rows: T(n,k) = number of simple graphs on n labeled nodes with k edges that are node-disjoint unions of undirected cycle subgraphs.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

Formula

A144209 Triangle T(n,k), n>=0, 0<=k<=n, read by rows: T(n,k) = number of simple graphs on n labeled nodes with k edges where each maximally connected subgraph consists of a single node or has a unique cycle of length 4.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A198148 a(n) = n*(n+2)*(9 - 7*(-1)^n)/16.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A234251 Triangle T(n, k) = Number of ways to choose k points from an n X n X n triangular grid so that no three of them form a 2 X 2 X 2 subtriangle. Triangle T read by rows.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A243214 Number of ways to place 5 points on a triangular grid of side n, so that no three of them are vertices of an equilateral triangle with sides parallel to the grid.

Views

Author

Keywords

A198148 a(n) = n(n+2)(9 - 7*(-1)^n)/16.