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

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, 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, 1, 15, 78, 165, 126, 21, 1, 16, 91, 220, 210, 56, 1, 1, 17, 105, 286, 330, 126, 7, 1, 18, 120

Offset

0,7

Comments

Row sums are A000930, antidiagonal sums are A003269.

Row n contains 1+floor(n/3) terms.

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

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

Links

Alois P. Heinz, Rows n = 0..250, flattened

Michael A. Allen, Connections between Combinations Without Specified Separations and Strongly Restricted Permutations, Compositions, and Bit Strings, arXiv:2409.00624 [math.CO], 2024. See p. 13.

Richard J. Mathar, Tiling n x m rectangles with 1 X 1 and s X s squares, arXiv:1609.03964 [math.CO], 2016, Section 4.2.

Richard J. Mathar, Bivariate Generating Functions Enumerating Non-Bonding Dominoes on Rectangular Boards, arXiv:2404.18806 [math.CO], 2024. See p. 7.

Michel Rigo, Manon Stipulanti, and Markus A. Whiteland, Gapped Binomial Complexities in Sequences, Univ. Liège (Belgium 2023).

Formula

T(n,k) = binomial(n-2k,k) (0 <= k <= n/3). - Emeric Deutsch, Aug 15 2010

G.f.: 1/(1 - x)/(1 - y*x^3/(1 - x)) = 1/(1 - x - y*x^3). - Geoffrey Critzer, Jun 25 2014

Example

Triangle begins:
  1;
  1;
  1;
  1, 1;
  1, 2;
  1, 3;
  1, 4, 1;
  1, 5, 3;

Maple

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

Mathematica

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 *)

Prog

(Magma) /* As triangle */ [[Binomial(n-2*k, k): k in [0..n div 3]]: n in [0.. 15]]; // Vincenzo Librandi, Jul 23 2019

Crossrefs

Cf. A007318, A011973, A003269, A000930 (row sums), A349862 (max row values).

Sequence in context: A142593 A360285 A226248 * A305556 A177812 A280504

Adjacent sequences: A102544 A102545 A102546 * A102548 A102549 A102550

Keyword

nonn,tabf

Author

Gerald McGarvey, Feb 24 2005

Status

approved