A093566 -id:A093566 - cp's OEIS frontend

A352472 Triangle T(n,k) read by rows: the number of traceless symmetric binary n X n matrices with 2k one's and no all-1 2 X 2 submatrix.

1, 1, 1, 1, 3, 3, 1, 1, 6, 15, 20, 12, 1, 10, 45, 120, 195, 162, 15, 1, 15, 105, 455, 1320, 2508, 2680, 900, 1, 21, 210, 1330, 5880, 18564, 40474, 54750, 35595, 6615, 1, 28, 378, 3276, 20265, 93240, 320040, 795120, 1333080, 1323840, 619920, 90720, 1, 36, 630, 7140, 58527, 364896

Offset: 1

R. J. Mathar, Mar 17 2022

Symmetry and traceless mean that the number of 1's is always even; the corresponding zeros for odd numbers are not shown here.

			The triangle starts at 1 X 1 matrices and 0,2,4,... ones as
1: 1;
2: 1  1;
3: 1  3   3    1;
4: 1  6  15   20    12;
5: 1 10  45  120   195    162      15;
6: 1 15 105  455  1320   2508    2680     900;
7: 1 21 210 1330  5880  18564   40474   54750    35595     6615;
8: 1 28 378 3276 20265  93240  320040  795120  1333080  1323840   619920    90720;
9: 1 36 630 7140 58527 364896 1763076 6578640 18514935 37535932 50808870 40684140 15892065 1995840;

Max Alekseyev, Table of n, a(n) for n = 1..205
Index entries for sequences of squarefree graphs

Cf. A350189 (allows nonzero trace), A345249 (row sums), A006855 (row lengths minus 1), A191966 (rightmost values).

T(n,1) = A000217(n-1). - R. J. Mathar, Mar 25 2022
T(n,2) = 3*A000332(n+1). T(n,3) = A093566(n+1). - Conjectured by R. J. Mathar, Mar 25 2022; proved by Max Alekseyev, Apr 02 2022
G.f.: F(x,y) = Sum_{n,k} T(n,k)*(x^n/n!)*y^k = exp( Sum_G x^n(G) * y^e(G) / |Aut(G)| ), where G runs over the connected squarefree graphs (cf. A077269), n(G) and e(G) are the numbers of nodes and edges in G, and Aut(G) is the automorphism group of G. It follows that F(x,y) = exp(x) * (1 + (1/2)*x^2*y + ((1/2)*x^3 + (1/8)*x^4)*y^2 + ((1/6)*x^3 + (2/3)*x^4 + (1/4)*x^5 + (1/48)*x^6)*y^3 + O(y^4)), implying the above formulas for T(n,2) and T(n,3). - Max Alekseyev, Apr 02 2022

A337959 Number of chiral pairs of colorings of the 30 triangular faces of a regular icosahedron or the 30 vertices of a regular dodecahedron using n or fewer colors.

Original entry on oeis.org

0, 8388, 28998090, 9160633008, 794699283870, 30467722237092, 664933856235516, 9607670743188672, 101313843935748516, 833333209516666980, 5606249568529546134, 31947998829845093424, 158374695227965468434

Offset: 1

Robert A. Russell, Oct 03 2020

Each member of a chiral pair is a reflection, but not a rotation, of the other. The Schläfli symbols for the regular icosahedron and regular dodecahedron are {3,5} and {5,3} respectively. They are mutually dual.

Index entries for linear recurrences with constant coefficients, signature (21, -210, 1330, -5985, 20349, -54264, 116280, -203490, 293930, -352716, 352716, -293930, 203490, -116280, 54264, -20349, 5985, -1330, 210, -21, 1).

Cf. A054472 (oriented), A252704 (unoriented), A337960 (achiral).
Other elements: A337964 (edges), A337961 (dodecahedron faces, icosahedron vertices).
Other polyhedra: A000332 (tetrahedron), A093566(n+1) (cube faces, octahedron vertices), A337896 (octahedron faces, cube vertices).

Table[(n^20-15n^12+14n^10+20n^8+4n^4-24n^2)/120,{n,30}]

a(n) = (n-1) * n^2 * (n+1) * (n^2+2) * (n^14 - n^12 + 3*n^10 - 5*n^8 - 4*n^6 + 8*n^4 + 4*n^2 + 12) /120.
a(n) = 8388*C(n,2) + 28972926*C(n,3) + 9044690976*C(n,4) + 749186015850*C(n,5) + 25836356193012*C(n,6) + 468028878138864*C(n,7) + 5097432576698784*C(n,8) + 36322117709159520*C(n,9) + 178947768558202560*C(n,10) + 632296225414909440*C(n,11) + 1640646875114311680*C(n,12) + 3168965153453299200*C(n,13) + 4578694359419980800*C(n,14) + 4929160839482880000*C(n,15) + 3897035952819609600*C(n,16) + 2197214626134528000*C(n,17) + 836310065310720000*C(n,18) + 192604742313984000*C(n,19) + 20274183401472000*C(n,20), where the coefficient of C(n,k) is the number of chiral pairs of colorings using exactly k colors.
a(n) = A054472(n) - A252704(n) = (A054472(n) - A337960(n)) / 2 = A252704(n) - A337960(n).

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).

A093567 Binomial (Binomial (n,2), 3) - Binomial (Binomial (n,3), 2).

Original entry on oeis.org

0, 1, 14, 75, 265, 735, 1736, 3654, 7050, 12705, 21670, 35321, 55419, 84175, 124320, 179180, 252756, 349809, 475950, 637735, 842765, 1099791, 1418824, 1811250, 2289950, 2869425, 3565926, 4397589, 5384575, 6549215, 7916160, 9512536

Offset: 2

Robert G. Wilson v and Santi Spadaro, Mar 31 2004

nonn

All terms are positive: A093566 >= A054563 ==> C( C(n,2), 3) >= C( C(n,3), 2) ==> n^2*(n^4 + 3n^3 -35n^2 + 69n -38)/144 >= 0 ==> (n - 2)(n - 1)(n^2 + 6n - 19) ==> 0 which it is for all n >= 2.

Solomon W. Golomb, Iterated binomial coefficients, Amer. Math. Monthly, 87 (1980), 719-727.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A054563, A093566.

A093567:=n->binomial(binomial(n, 2), 3) - binomial(binomial(n, 3), 2); seq(A093567(n), n=2..30); # Wesley Ivan Hurt, Feb 02 2014

Table[ Binomial[ Binomial[n, 2], 3] - Binomial[ Binomial[n, 3], 2], {n, 2, 34}]
LinearRecurrence[{7,-21,35,-35,21,-7,1},{0,1,14,75,265,735,1736},40] (* Harvey P. Dale, Jun 12 2016 *)

a(n) = binomial(binomial(n,2), 3) - binomial(binomial(n,3), 2); \\ Michel Marcus, Oct 01 2017

a(n) = A093566(n) - A054563(n).

G.f.: x^3*(-1-7*x+2*x^2+x^3)/(x-1)^7. - R. J. Mathar, Dec 08 2010

A175991 a(n) = binomial(binomial(binomial(n, 2), 3), 4)/5.

Original entry on oeis.org

969, 1642914, 352470391, 25957590316, 958073067315, 21639468423573, 337726148030733, 3946787095970862, 36534727415378192, 279109860906071195, 1815047255456722287, 10290566991057546557, 51837653320551263438, 235568544405588437778, 977816056476957297015, 3745739023587032569461, 13356862465688668653111

Offset: 4

Roger L. Bagula, Dec 06 2010

nonn

Cf. A093566.

Table[Binomial[Binomial[Binomial[n, 2], 3], 4]/5, {n, 4, 30}]

From R. J. Mathar, Dec 08 2010: (Start)
a(n) = binomial(A093566(n+1),4)/5.
a(n) = n *(n-1) *(n-2) *(n-3) *(n+2) *(n+1) *(n^2-n-4) *(n^6-3*n^5-3*n^4+11*n^3+2*n^2-8*n-96) *(n^4-2*n^3+n^2+8) *(n^6-3*n^5-3*n^4+11*n^3+2*n^2-8*n-144) /637009920. (End)

A176018 a(n) = binomial(A000460(n), 3).

Original entry on oeis.org

165, 45760, 4545100, 280859635, 13177343466, 519435748656, 18247149400480, 592679189880470, 18233421432967455, 539997542625453900, 15568435368162197664, 440371777466788015089, 12288775148056292092340, 339634237637121659008140

Offset: 4

Roger L. Bagula, Dec 06 2010

nonn

Cf. A093566.

Table[Binomial[Eulerian[n + 1, 2], 3], {n, 3, 30}]

cp's OEIS Frontend

A352472 Triangle T(n,k) read by rows: the number of traceless symmetric binary n X n matrices with 2k one's and no all-1 2 X 2 submatrix.

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A337959 Number of chiral pairs of colorings of the 30 triangular faces of a regular icosahedron or the 30 vertices of a regular dodecahedron using n or fewer colors.

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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.

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A093567 Binomial (Binomial (n,2), 3) - Binomial (Binomial (n,3), 2).

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A175991 a(n) = binomial(binomial(binomial(n, 2), 3), 4)/5.

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A176018 a(n) = binomial(A000460(n), 3).

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