A001752 -id:A001752 - cp's OEIS frontend

A362903 Array read by antidiagonals: T(n,k) is the number of nonisomorphic k-tuples of involutions on a (2n)-set that pairwise commute.

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 8, 11, 4, 1, 1, 16, 43, 24, 5, 1, 1, 32, 171, 176, 46, 6, 1, 1, 64, 683, 1376, 611, 80, 7, 1, 1, 128, 2731, 10944, 9281, 1864, 130, 8, 1, 1, 256, 10923, 87424, 146445, 54384, 5161, 200, 9, 1, 1, 512, 43691, 699136, 2334181, 1696352, 285939, 13184, 295, 10, 1

Offset: 0

Andrew Howroyd, May 11 2023

Two involutions x,y commute if x*y = y*x. Isomorphism is up to permutation of the elements of the (2n)-set. T(n,k) also gives the values for a (2n+1)-set.

			Array begins:
======================================================
n/k| 0 1   2     3       4          5            6 ...
---+--------------------------------------------------
0  | 1 1   1     1       1          1            1 ...
1  | 1 2   4     8      16         32           64 ...
2  | 1 3  11    43     171        683         2731 ...
3  | 1 4  24   176    1376      10944        87424 ...
4  | 1 5  46   611    9281     146445      2334181 ...
5  | 1 6  80  1864   54384    1696352     53885632 ...
6  | 1 7 130  5161  285939   17562679   1110290303 ...
7  | 1 8 200 13184 1372224  165343616  20774749952 ...
8  | 1 9 295 31532 6101080 1436647664 358238974304 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals)

Columns k=0..3 are A000012, A000027(n-1), A001752, A362904.
Rows n=1..3 are A000079, A007583, A103334(n+1).
Cf. A022166, A362648, A362824, A362826.

\\ B(n, k) is A022166.
B(n, k)={polcoef(x^k/prod(j=0, k, 1-2^j*x + O(x*x^n)), n)}
C(k,n) = Vec(1/prod(j=0, min(k-1, logint(n, 2)), (1 - x^(2^j) + O(x*x^n))^B(k,j+1), 1 - x + O(x*x^n)))
M(n,m=n) = Mat(vector(m+1, k, C(k-1, n)~))
{ my(A=M(7)); for(i=1, #A, print(A[i,])) }

G.f. of column k: 1/((1 - x)*Product_{j=0..k-1} (1 - x^(2^j))^A022166(k,j+1)).

A057971 Number of 5 x n binary matrices with 2 unit columns up to row and column permutations.

Original entry on oeis.org

2, 18, 133, 873, 5182, 27786, 135370, 602454, 2466628, 9358497, 33134431, 110184932, 346141949, 1032550097, 2938104492, 8006865684, 20971632456, 52958252851, 129291697111, 305924724070, 703108665327, 1572722761341

Offset: 2

Vladeta Jovovic, Oct 21 2000

nonn

A unit column of a binary matrix is a column with only one 1. First differences of a(n) give number of minimal 5 - covers of an unlabeled n - set that cover 7 points of that set uniquely (if offset is 7).

Vladeta Jovovic, Number of minimal covers of an unlabeled n - set that cover k points of that set uniquely
Vladeta Jovovic, Number of binary matrices with fixed number of unit columns up to row and column permutations

Cf. A001752, A056885, A057222, A057223, A057524, A057669, A057963-A057968, A057970-A057972, A057969, A057970, A057972.

Number of 5 x n binary matrices with k unit columns up to row and column permutations is coefficient of x^k in (1/5!)*(Z(S_n; 27 + 5*x, 27 + 5*x^2, ...) + 10*Z(S_n; 13 + 3*x, 27 + 5*x^2, 13 + 3*x^3, 27 + 5*x^4, ...) + 15*Z(S_n; 7 + x, 27 + 5*x^2, 7 + x^3, 27 + 5*x^4, ...) + 20*Z(S_n; 6 + 2*x, 6 + 2*x^2, 27 + 5*x^3, 6 + 2*x^4, 6 + 2*x^5, 27 + 5*x^6, ...) +
20*Z(S_n; 4, 6 + 2*x^2, 13 + 3*x^3, 6 + 2*x^4, 4, 27 + 5*x^6, 4, 6 + 2*x^8, 13 + 3*x^9, 6 + 2*x^10, 4, 27 + 5*x^12, ...) + 30*Z(S_n; 3 + x, 7 + x^2, 3 + x^3, 27 + 5*x^4, 3 + x^5, 7 + x^6, 3 + x^7, 27 + 5*x^8, ...) + 24*Z(S_n; 2, 2, 2, 2, 27 + 5*x^5, 2, 2, 2, 2, 27 + 5*x^10, ...)),
where Z(S_n; x_1, x_2, ..., x_n) is cycle index of symmetric group S_n of degree n.
G.f.: x^2/120*(15/(1 - x^1)^27 + 70/(1 - x^1)^13/(1 - x^2)^7 + 45/(1 - x^1)^7/(1 - x^2)^10 + 60/(1 - x^1)^6/(1 - x^3)^7 + 20/(1 - x^1)^4/(1 - x^2)^1/(1 - x^3)^3/(1 - x^6)^2 + 30/(1 - x^1)^3/(1 - x^2)^2/(1 - x^4)^5).

A301740 The number of trees with 5 nodes labeled by positive integers, where each tree's label sum is n.

Original entry on oeis.org

3, 9, 24, 50, 96, 164, 267, 408, 603, 856, 1186, 1598, 2115, 2742, 3505, 4411, 5489, 6746, 8215, 9904, 11849, 14059, 16573, 19401, 22586, 26138, 30103, 34493, 39357, 44707, 50596, 57037, 64086, 71757, 80109, 89157, 98964, 109545, 120966, 133244, 146448, 160595, 175758, 191955

Offset: 5

R. J. Mathar, Mar 26 2018

Computed by the sum over the A000055(5)=3 shapes of the trees: the linear graph of the n-Pentane, the branched 2-Methyl-Butane, and the star graph of (1,1)-Bimethyl-Propane.

			a(5)=3 because there is a linear tree with all labels equal 1, the branched tree with all labels equal to 1, and the star tree with all labels equal 1.

R. J. Mathar, Labeled Trees with Fixed Node Label sum, sequence v_5.
Index entries for linear recurrences with constant coefficients, signature (2,0,-1,0,-2,2,0,1,0,-2,1).

Cf. A002620 (labeled trees with 3 nodes), A301739 (labeled trees with 4 nodes).

-x^5*(3+3*x+6*x^2+5*x^3+5*x^4+2*x^5+x^6)/(1+x^2)/(1+x+x^2)/(1+x)^2/(x-1)^5 ;
taylor(%,x=0,80) ;
gfun[seriestolist](%) ;

LinearRecurrence[{2,0,-1,0,-2,2,0,1,0,-2,1},{3,9,24,50,96,164,267,408,603,856,1186},50] (* Harvey P. Dale, Jun 16 2025 *)

a(n) = A005994(n-5)+A001752(n-5)+A002621(n-5).

A204185 Number of quadrilaterals in a triangular matchstick arrangement of side n.

Original entry on oeis.org

0, 0, 6, 33, 102, 243, 492, 894, 1500, 2370, 3570, 5175, 7266, 9933, 13272, 17388, 22392, 28404, 35550, 43965, 53790, 65175, 78276, 93258, 110292, 129558, 151242, 175539, 202650, 232785, 266160, 303000, 343536, 388008, 436662, 489753, 547542, 610299, 678300, 751830, 831180, 916650, 1008546, 1107183, 1212882, 1325973, 1446792, 1575684, 1713000, 1859100, 2014350

Offset: 0

Elliott Line & Paul Bostock (enigma.mensa(AT)yahoo.co.uk), Jan 12 2012

The total number of parallelograms and trapezoids that appear in a triangular matchstick array of side n.
Can always be split into three equal sets, parallelograms 'belonging' to the side of the triangle that none of its sides are parallel to, and trapezoids 'belonging' to the side of the triangle that two of its sides are parallel to.
Rhombuses belonging to each side are A173196(n).
Irregular parallelograms belonging to each side are 2*A001752(n-3).
'Upside down' trapezoids (those where the shorter of the two parallel sides is closest to the parallel side of the triangle) belonging to each side are A001752(n-3).
'Right side up' trapezoids belonging to each side are A000332(n+2).

			a(3) = 33 because the following figure contains 33 quadrilaterals (15 parallelograms and 18 trapezoids)
....... /\
...... /\/\
..... /\/\/\
Size and quantity of each quadrilateral in above figure:
2 triangles: 9
3 triangles: 12
4 triangles: 6
5 triangles: 3
8 triangles: 3

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,-5,0,5,-4,1).

Cf. A173196 = number of rhombuses of a particular orientation; A001752, related to number of irregular parallelograms and number of 'upside down' trapezoids; A000332, related to number of 'right side up' trapezoids (see comments above); A002717 = number of triangles in a triangular matchstick arrangement; A000217 = triangular numbers.

nxt[{n_,a_}]:={n+1,a+Floor[n(n+2) (10(n+1)-3)/8]}; Transpose[ NestList[ nxt,{0,0},50]][[2]] (* Harvey P. Dale, Jan 11 2013 *)

concat([0,0], Vec(-3*x^2*(3*x+2)/((x-1)^5*(x+1)) + O(x^100))) \\ Colin Barker, Mar 16 2015

a(n) = Sum_{k=1..n-1} b(k)*T(n-k), where b(m) = 3*floor(5*m/2) and T(m) is the m-th triangular number A000217.
a(n) = a(n-1) + floor((n+1)*(n-1)*(10*n-3)/8).
a(n) = 3*(A173196(n) + A000332(n+2) + 3*A001752(n-3)) (see comments above).
From Colin Barker, Mar 16 2015: (Start)
a(n) = (3-3*(-1)^n-16*n-16*n^2+16*n^3+10*n^4)/32.
a(n) = 4*a(n-1)-5*a(n-2)+5*a(n-4)-4*a(n-5)+a(n-6).
G.f.: -3*x^2*(3*x+2) / ((x-1)^5*(x+1)). (End)
E.g.f.: (x*(5*x^3 + 38*x^2 + 51*x - 3)*cosh(x) + (5*x^4 + 38*x^3 + 51*x^2 - 3*x + 3)*sinh(x))/16. - Stefano Spezia, Jul 19 2022

A334891 Number of ways to choose 4 points that form a square from the A000292(n) points in a regular tetrahedral grid where each side has n vertices.

Original entry on oeis.org

0, 0, 3, 12, 36, 84, 174, 336, 612, 1044, 1701

Offset: 0

Peter Kagey, May 14 2020

a(n) >= 3*A001752(n-2).

			For n = 4, three of the a(4) = 36 squares are (in barycentric coordinates)
  {(0,2,1,1),(1,1,0,2),(1,1,2,0),(2,0,1,1)},
  {(0,0,2,2),(0,2,0,2),(2,0,2,0),(2,2,0,0)}, and
  {(0,0,1,3),(0,1,0,3),(1,0,1,2),(1,1,0,2)}.
The other squares can be derived from these by translations or symmetries of the tetrahedron.

Cf. A000292, A001752.

Cf. A334581 (equilateral triangle), A334881 (cubic grid).

cp's OEIS Frontend

A362903 Array read by antidiagonals: T(n,k) is the number of nonisomorphic k-tuples of involutions on a (2n)-set that pairwise commute.

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A057971 Number of 5 x n binary matrices with 2 unit columns up to row and column permutations.

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A301740 The number of trees with 5 nodes labeled by positive integers, where each tree's label sum is n.

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A204185 Number of quadrilaterals in a triangular matchstick arrangement of side n.

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Mathematica

PARI

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A334891 Number of ways to choose 4 points that form a square from the A000292(n) points in a regular tetrahedral grid where each side has n vertices.

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