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A102547
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Triangle read by rows, formed from antidiagonals of the antidiagonals (A011973) of Pascal's triangle (A007318).
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7
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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
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OFFSET
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0,7
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COMMENTS
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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
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LINKS
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FORMULA
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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
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EXAMPLE
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Triangle begins:
1;
1;
1;
1, 1;
1, 2;
1, 3;
1, 4, 1;
1, 5, 3;
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(Magma) /* As triangle */ [[Binomial(n-2*k, k): k in [0..n div 3]]: n in [0.. 15]]; // Vincenzo Librandi, Jul 23 2019
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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