cp's OEIS Frontend

A102547 Triangle read by rows, formed from antidiagonals of the antidiagonals (A011973) of Pascal's triangle (A007318).

%I A102547 #59 Sep 11 2024 00:40:13
%S A102547 1,1,1,1,1,1,2,1,3,1,4,1,1,5,3,1,6,6,1,7,10,1,1,8,15,4,1,9,21,10,1,10,
%T A102547 28,20,1,1,11,36,35,5,1,12,45,56,15,1,13,55,84,35,1,1,14,66,120,70,6,
%U A102547 1,15,78,165,126,21,1,16,91,220,210,56,1,1,17,105,286,330,126,7,1,18,120
%N A102547 Triangle read by rows, formed from antidiagonals of the antidiagonals (A011973) of Pascal's triangle (A007318).
%C A102547 Row sums are A000930, antidiagonal sums are A003269.
%C A102547 Row n contains 1+floor(n/3) terms.
%C A102547 T(n,k) is the number of compositions of n+3 with k+1 parts, all at least 3. Example: T(9,2) = binomial(5,2) = 10 because we have 336, 363, 633, 345, 354, 435, 453, 534, 543, and 444. - _Emeric Deutsch_, Aug 15 2010
%C A102547 T(n+2,k) is the number of k-subsets of {1..n} with values at least 3 apart. For example, T(7,2) = 3 corresponds to the subsets {1,4},{1,5},{2,5} of {1..5}. - _Enrique Navarrete_, Dec 19 2021
%H A102547 Alois P. Heinz, <a href="/A102547/b102547.txt">Rows n = 0..250, flattened</a>
%H A102547 Michael A. Allen, <a href="https://arxiv.org/abs/2409.00624">Connections between Combinations Without Specified Separations and Strongly Restricted Permutations, Compositions, and Bit Strings</a>, arXiv:2409.00624 [math.CO], 2024. See p. 13.
%H A102547 Richard J. Mathar, <a href="http://arxiv.org/abs/1609.03964">Tiling n x m rectangles with 1 X 1 and s X s squares</a>, arXiv:1609.03964 [math.CO], 2016, Section 4.2.
%H A102547 Richard J. Mathar, <a href="https://arxiv.org/abs/2404.18806">Bivariate Generating Functions Enumerating Non-Bonding Dominoes on Rectangular Boards</a>, arXiv:2404.18806 [math.CO], 2024. See p. 7.
%H A102547 Michel Rigo, Manon Stipulanti, and Markus A. Whiteland, <a href="https://orbi.uliege.be/bitstream/2268/302524/1/isit2023.pdf">Gapped Binomial Complexities in Sequences</a>, Univ. Liège (Belgium 2023).
%F A102547 T(n,k) = binomial(n-2k,k) (0 <= k <= n/3). - _Emeric Deutsch_, Aug 15 2010
%F A102547 G.f.: 1/(1 - x)/(1 - y*x^3/(1 - x)) = 1/(1 - x - y*x^3). - _Geoffrey Critzer_, Jun 25 2014
%e A102547 Triangle begins:
%e A102547   1;
%e A102547   1;
%e A102547   1;
%e A102547   1, 1;
%e A102547   1, 2;
%e A102547   1, 3;
%e A102547   1, 4, 1;
%e A102547   1, 5, 3;
%p A102547 for n from 0 to 20 do seq(binomial(n-2*k, k), k = 0 .. floor((1/3)*n)) end do; # yields sequence in triangular form. - _Emeric Deutsch_, Aug 15 2010
%t A102547 nn=20;Map[Select[#,#>0&]&,CoefficientList[Series[1/(1-x)/(1-y x^3/(1-x)),{x,0,nn}],{x,y}]]//Grid (* _Geoffrey Critzer_, Jun 25 2014 *)
%o A102547 (Magma) /* As triangle */ [[Binomial(n-2*k,k): k in [0..n div 3]]: n in [0.. 15]]; // _Vincenzo Librandi_, Jul 23 2019
%Y A102547 Cf. A007318, A011973, A003269, A000930 (row sums), A349862 (max row values).
%K A102547 nonn,tabf
%O A102547 0,7
%A A102547 _Gerald McGarvey_, Feb 24 2005