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 A216652 Triangular array read by rows: T(n,k) is the number of compositions of n into exactly k distinct parts. 10
 1, 1, 1, 2, 1, 2, 1, 4, 1, 4, 6, 1, 6, 6, 1, 6, 12, 1, 8, 18, 1, 8, 24, 24, 1, 10, 30, 24, 1, 10, 42, 48, 1, 12, 48, 72, 1, 12, 60, 120, 1, 14, 72, 144, 120, 1, 14, 84, 216, 120, 1, 16, 96, 264, 240, 1, 16, 114, 360, 360, 1, 18, 126, 432, 600, 1, 18, 144, 552, 840 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS Same as A072574, with zeros dropped. [Joerg Arndt, Oct 20 2012] Row sums = A032020. Row n contains A003056(n) = floor((sqrt(8*n+1)-1)/2) terms (number of terms increases by one at each triangular number). LINKS Alois P. Heinz, Rows n = 1..500, flattened B. Richmond and A. Knopfmacher, Compositions with distinct parts, Aequationes Mathematicae 49 (1995), pp. 86-97. FORMULA G.f.: Sum_{i>=0} Product_{j=1..i} y*j*x^j/(1-x^j). T(n,k) = A008289(n,k)*k!. EXAMPLE Triangle starts: [ 1]  1; [ 2]  1; [ 3]  1, 2; [ 4]  1, 2; [ 5]  1, 4; [ 6]  1, 4, 6; [ 7]  1, 6, 6; [ 8]  1, 6, 12; [ 9]  1, 8, 18; [10]  1, 8, 24, 24; [11]  1, 10, 30, 24; [12]  1, 10, 42, 48; [13]  1, 12, 48, 72; [14]  1, 12, 60, 120; [15]  1, 14, 72, 144, 120; [16]  1, 14, 84, 216, 120; [17]  1, 16, 96, 264, 240; [18]  1, 16, 114, 360, 360; [19]  1, 18, 126, 432, 600; [20]  1, 18, 144, 552, 840; T(5,2) = 4 because we have: 4+1, 1+4, 3+2, 2+3. MAPLE b:= proc(n, k) option remember; `if`(n<0, 0, `if`(n=0, 1,       `if`(k<1, 0, b(n, k-1) +b(n-k, k))))     end: T:= (n, k)-> b(n-k*(k+1)/2, k)*k!: seq(seq(T(n, k), k=1..floor((sqrt(8*n+1)-1)/2)), n=1..24);  # Alois P. Heinz, Sep 12 2012 MATHEMATICA nn=20; f[list_]:=Select[list, #>0&]; Map[f, Drop[CoefficientList[Series[ Sum[Product[j y x^j/(1-x^j), {j, 1, k}], {k, 0, nn}], {x, 0, nn}], {x, y}], 1]]//Flatten CROSSREFS Cf. A003056, A008289, A072574, A097910. Sequence in context: A343411 A287477 A231473 * A331980 A055684 A300584 Adjacent sequences:  A216649 A216650 A216651 * A216653 A216654 A216655 KEYWORD nonn,tabf AUTHOR Geoffrey Critzer, Sep 12 2012 STATUS approved

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Last modified July 29 17:41 EDT 2021. Contains 346346 sequences. (Running on oeis4.)