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%I #97 Jan 19 2024 04:33:05
%S 1,4,11,24,46,80,130,200,295,420,581,784,1036,1344,1716,2160,2685,
%T 3300,4015,4840,5786,6864,8086,9464,11011,12740,14665,16800,19160,
%U 21760,24616,27744,31161,34884,38931,43320,48070,53200,58730,64680,71071,77924,85261
%N Expansion of 1/((1+x)*(1-x)^5).
%C 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
%C 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.
%C 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).
%C Transform of tetrahedral numbers, binomial(n+3,3), under the Riordan array (1/(1-x^2), x). - _Paul Barry_, Apr 16 2005
%C Equals triangle A152205 as an infinite lower triangular matrix * [1, 2, 3, ...]. - _Gary W. Adamson_, Feb 14 2010
%C 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
%C 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
%C 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
%C 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
%D T. A. Saaty, The Minimum Number of Intersections in Complete Graphs, Proc. Natl. Acad. Sci. USA., 52 (1964), 688-690.
%H Vincenzo Librandi, <a href="/A001752/b001752.txt">Table of n, a(n) for n = 0..10000</a>
%H Dragomir Z. Djokovic, <a href="http://arXiv.org/abs/math.AC/0609262">Poincaré series of some pure and mixed trace algebras of two generic matrices</a>, arXiv:math/0609262 [math.AC], 2006. See Table 4.
%H Jia Huang, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Huang/huang8.html">Partially Palindromic Compositions</a>, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See pp. 4, 17.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,0,5,-4,1).
%F a(n) = floor(((n+3)^2 - 1)*((n+3)^2 - 3)/48).
%F G.f.: 1/((1+x)*(1-x)^5).
%F a(n) - 2*a(n-1) + a(n-2) = A002620(n+2).
%F a(n) + a(n-1) = A000332(n+4).
%F a(n) - a(n-2) = A000292(n+1).
%F a(n) = Sum_{k=0..n} (-1)^(n-k)*C(k+4, 4). - _Paul Barry_, Jul 01 2003
%F 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
%F From _Paul Barry_, Apr 16 2005: (Start)
%F a(n) = Sum_{k=0..floor(n/2)} binomial(n-2k+3, 3).
%F a(n) = Sum_{k=0..n} binomial(k+3, 3)*(1-(-1)^(n+k-1))/2. (End)
%F a(n) = A108561(n+5,n) for n > 0. - _Reinhard Zumkeller_, Jun 10 2005
%F From _Wesley Ivan Hurt_, Apr 01 2015: (Start)
%F a(n) = 4*a(n-1) - 5*a(n-2) + 5*(n-4) - 4*a(n-5) + a(n-6).
%F a(n) = Sum_{i=0..n+3} (n+3-i) * floor(i^2/2)/2. (End)
%F 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
%F Convolution of A000217 and A004526. - _R. J. Mathar_, Mar 29 2018
%F 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
%e There are 4 binary 3 X 2 matrices with 1 unit column up to row and column permutations:
%e [0 0] [0 0] [0 1] [0 1]
%e [0 0] [0 1] [0 1] [0 1]
%e [0 1] [1 1] [1 0] [1 1].
%e 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
%p A001752:=n->(3*(-1)^n+93+168*n+100*n^2+24*n^3+2*n^4)/96:
%p seq(A001752(n), n=0..50); # _Wesley Ivan Hurt_, Apr 01 2015
%t a = {1, 4}; Do[AppendTo[a, a[[n - 2]] + (n*(n + 1)*(n + 2))/6], {n, 3, 10}]; a
%t (* Number of octahedra *) nnn = 100; Teo[n_] := (n - 1) n (n + 1)/6
%t Table[Sum[Teo[n - nn], {nn, 0, n - 1, 2}], {n, 1, nnn}]
%t (* _V.J. Pohjola_, Sep 13 2012 *)
%t LinearRecurrence[{4,-5,0,5,-4,1},{1,4,11,24,46,80},50] (* _Harvey P. Dale_, Feb 07 2019 *)
%o (PARI) a(n)=if(n<0,0,((n+3)^2-1)*((n+3)^2-3)/48-if(n%2,1/16))
%o (PARI) a(n)=(n^4+12*n^3+50*n^2+84*n+46)\/48 \\ _Charles R Greathouse IV_, Sep 13 2012
%o (Magma) [Floor(((n+3)^2-1)*((n+3)^2-3)/48): n in [0..40]]; // _Vincenzo Librandi_, Aug 15 2011
%Y Cf. A000217, A000292, A000332, A000929, A002620, A004526, A010751, A056885, A057524, A108561, A152205, A216172, A216173, A216175.
%Y Cf. A001753 (partial sums), A002623 (first differences), A158454 (signed column k=2), A169792 (binomial transform).
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
%E Formulae corrected by _Bruno Berselli_, Sep 13 2012
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