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A001752 Expansion of 1/((1+x)*(1-x)^5). 34
1, 4, 11, 24, 46, 80, 130, 200, 295, 420, 581, 784, 1036, 1344, 1716, 2160, 2685, 3300, 4015, 4840, 5786, 6864, 8086, 9464, 11011, 12740, 14665, 16800, 19160, 21760, 24616, 27744, 31161, 34884, 38931, 43320, 48070, 53200, 58730, 64680, 71071, 77924, 85261 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Define a unit column of a binary matrix to be a column with only one 1. a(n) = number of 3 X n binary matrices with 1 unit column up to row and column permutations (if offset is 1). - Vladeta Jovovic, Sep 09 2000

Generally, number of 3 X n binary matrices with k=0,1,2,... unit columns, up to row and column permutations, is the coefficient of x^k in 1/6*(Z(S_n; 5 + 3*x,5 + 3*x^2, ...) + 3*Z(S_n; 3 + x,5 + 3*x^2,3 + x^3,5 + 3*x^4, ...) + 2*Z(S_n; 2,2,5 + 3*x^3,2,2,5 + 3*x^6, ...)), where Z(S_n; x_1,x_2,...,x_n) is the cycle index of symmetric group S_n of degree n.

First differences of a(n) give number of minimal 3-covers of an unlabeled n-set that cover 4 points of that set uniquely (if offset is 4).

Transform of tetrahedral numbers, binomial(n+3,3), under the Riordan array (1/(1-x^2), x). - Paul Barry, Apr 16 2005

Equals triangle A152205 as an infinite lower triangular matrix * [1, 2, 3, ...]. - Gary W. Adamson, Feb 14 2010

With a leading zero, number of all possible octahedra of any size, formed when intersecting a regular tetrahedron by planes parallel to its sides and dividing its edges into n equal parts. - V.J. Pohjola, Sep 13 2012

With 2 leading zeros and offset 1, the sequence becomes 0,0,1,4,11,... for n=1,2,3,... Call this b(n). Consider the partitions of n into two parts (p,q) with p <= q. Then b(n) is the total volume of the family of rectangular prisms with dimensions p, |q - p| and |q - p|. - Wesley Ivan Hurt, Apr 14 2018

Conjecture: For n > 2, a(n-3) is the absolute value of the coefficient of the term [x^(n-2)] in the characteristic polynomial of an n X n square matrix M(n) defined as the n-th principal submatrix of the array A010751 whose general element is given by M[i,j] = floor((j - i + 1)/2). - Stefano Spezia, Jan 12 2022

Consider the following drawing of the complete graph on n vertices K_n: Vertices 1, 2, ..., n are on a straight line. Any pair of nonconsecutive vertices (i, j) with i < j is connected by a semicircle that goes above the line if i is odd, and below if i is even. With four leading zeros and offset 1, a(n) gives the number of edge crossings of the aforementioned drawing of K_n. - Carlo Francisco E. Adajar, Mar 17 2022

REFERENCES

T. A. Saaty, The Minimum Number of Intersections in Complete Graphs, Proc. Natl. Acad. Sci. USA., 52 (1964), 688-690.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..10000

Dragomir Z. Djokovic, Poincaré series [or Poincare series] of some pure and mixed trace algebras of two generic matrices, arXiv:math/0609262 [math.AC], 2006. See Table 4.

Index entries for linear recurrences with constant coefficients, signature (4,-5,0,5,-4,1).

FORMULA

a(n) = floor(((n+3)^2 - 1)*((n+3)^2 - 3)/48).

G.f.: 1/((1+x)*(1-x)^5).

a(n) - 2*a(n-1) + a(n-2) = A002620(n+2).

a(n) + a(n-1) = A000332(n+4).

a(n) - a(n-2) = A000292(n+1).

a(n) = Sum_{k=0..n} (-1)^(n-k)*C(k+4, 4). - Paul Barry, Jul 01 2003

a(n) = (3*(-1)^n + 93 + 168*n + 100*n^2 + 24*n^3 + 2*n^4)/96. - Cecilia Rossiter (cecilia(AT)noticingnumbers.net), Dec 14 2004

From Paul Barry, Apr 16 2005: (Start)

a(n) = Sum_{k=0..floor(n/2)} binomial(n-2k+3, 3).

a(n) = Sum_{k=0..n} binomial(k+3, 3)*(1-(-1)^(n+k-1))/2. (End)

a(n) = A108561(n+5,n) for n > 0. - Reinhard Zumkeller, Jun 10 2005

From Wesley Ivan Hurt, Apr 01 2015: (Start)

a(n) = 4*a(n-1) - 5*a(n-2) + 5*(n-4) - 4*a(n-5) + a(n-6).

a(n) = Sum_{i=0..n+3} (n+3-i) * floor(i^2/2)/2. (End)

Boas-Buck recurrence: a(n) = (1/n)*Sum_{p=0..n-1} (5 + (-1)^(n-p))*a(p), n >= 1, a(0) = 1. See the Boas-Buck comment in A158454 (here for the unsigned column k = 2 with offset 0). - Wolfdieter Lang, Aug 10 2017

Convolution of A000217 and A004526. - R. J. Mathar, Mar 29 2018

E.g.f.: ((48 + 147*x + 93*x^2 + 18*x^3 + x^4)*cosh(x) + (45 + 147*x + 93*x^2 + 18*x^3 + x^4)*sinh(x))/48. - Stefano Spezia, Jan 12 2022

EXAMPLE

There are 4 binary 3 X 2 matrices with 1 unit column up to row and column permutations:

  [0 0] [0 0] [0 1] [0 1]

  [0 0] [0 1] [0 1] [0 1]

  [0 1] [1 1] [1 0] [1 1].

For n=5, the numbers of the octahedra, starting from the smallest size, are Te(5)=35, Te(3)=10, and Te(1)=1, the sum being 46. Te denotes the tetrahedral number A000292. - V.J. Pohjola, Sep 13 2012

MAPLE

A001752:=n->(3*(-1)^n+93+168*n+100*n^2+24*n^3+2*n^4)/96:

seq(A001752(n), n=0..50); # Wesley Ivan Hurt, Apr 01 2015

MATHEMATICA

a = {1, 4}; Do[AppendTo[a, a[[n - 2]] + (n*(n + 1)*(n + 2))/6], {n, 3, 10}]; a

(* Number of octahedra *) nnn = 100; Teo[n_] := (n - 1) n (n + 1)/6

Table[Sum[Teo[n - nn], {nn, 0, n - 1, 2}], {n, 1, nnn}]

(* V.J. Pohjola, Sep 13 2012 *)

LinearRecurrence[{4, -5, 0, 5, -4, 1}, {1, 4, 11, 24, 46, 80}, 50] (* Harvey P. Dale, Feb 07 2019 *)

PROG

(PARI) a(n)=if(n<0, 0, ((n+3)^2-1)*((n+3)^2-3)/48-if(n%2, 1/16))

(PARI) a(n)=(n^4+12*n^3+50*n^2+84*n+46)\/48 \\ Charles R Greathouse IV, Sep 13 2012

(Magma) [Floor(((n+3)^2-1)*((n+3)^2-3)/48): n in [0..40]]; // Vincenzo Librandi, Aug 15 2011

CROSSREFS

Cf. A000217, A000292, A000332, A000929, A002620, A004526, A010751, A056885, A057524, A108561, A152205, A216172, A216173, A216175.

Cf. A001753 (partial sums), A002623 (first differences), A158454 (signed column k=2).

Sequence in context: A006527 A167875 A057304 * A160860 A192748 A143075

Adjacent sequences:  A001749 A001750 A001751 * A001753 A001754 A001755

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

Formulae corrected by Bruno Berselli, Sep 13 2012

STATUS

approved

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Last modified July 3 04:45 EDT 2022. Contains 355030 sequences. (Running on oeis4.)