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A001752 Expansion of 1/((1+x)*(1-x)^5). 32
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

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

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. A057524, A056885, A152205, A000929, A158454 (signed column k=2), A216172, A216173, A216175, A001753 (partial sums), A002623 (first differences).

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 December 12 15:11 EST 2019. Contains 329960 sequences. (Running on oeis4.)