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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.

%I A216652 #17 Jul 24 2020 21:59:50
%S A216652 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,
%T A216652 42,48,1,12,48,72,1,12,60,120,1,14,72,144,120,1,14,84,216,120,1,16,96,
%U A216652 264,240,1,16,114,360,360,1,18,126,432,600,1,18,144,552,840
%N A216652 Triangular array read by rows: T(n,k) is the number of compositions of n into exactly k distinct parts.
%C A216652 Same as A072574, with zeros dropped. [_Joerg Arndt_, Oct 20 2012]
%C A216652 Row sums = A032020.
%C A216652 Row n contains A003056(n) = floor((sqrt(8*n+1)-1)/2) terms (number of terms increases by one at each triangular number).
%H A216652 Alois P. Heinz, <a href="/A216652/b216652.txt">Rows n = 1..500, flattened</a>
%H A216652 B. Richmond and A. Knopfmacher, <a href="http://dx.doi.org/10.1007/BF01827930">Compositions with distinct parts</a>, Aequationes Mathematicae 49 (1995), pp. 86-97.
%F A216652 G.f.: Sum_{i>=0} Product_{j=1..i} y*j*x^j/(1-x^j).
%F A216652 T(n,k) = A008289(n,k)*k!.
%e A216652 Triangle starts:
%e A216652 [ 1]  1;
%e A216652 [ 2]  1;
%e A216652 [ 3]  1, 2;
%e A216652 [ 4]  1, 2;
%e A216652 [ 5]  1, 4;
%e A216652 [ 6]  1, 4, 6;
%e A216652 [ 7]  1, 6, 6;
%e A216652 [ 8]  1, 6, 12;
%e A216652 [ 9]  1, 8, 18;
%e A216652 [10]  1, 8, 24, 24;
%e A216652 [11]  1, 10, 30, 24;
%e A216652 [12]  1, 10, 42, 48;
%e A216652 [13]  1, 12, 48, 72;
%e A216652 [14]  1, 12, 60, 120;
%e A216652 [15]  1, 14, 72, 144, 120;
%e A216652 [16]  1, 14, 84, 216, 120;
%e A216652 [17]  1, 16, 96, 264, 240;
%e A216652 [18]  1, 16, 114, 360, 360;
%e A216652 [19]  1, 18, 126, 432, 600;
%e A216652 [20]  1, 18, 144, 552, 840;
%e A216652 T(5,2) = 4 because we have: 4+1, 1+4, 3+2, 2+3.
%p A216652 b:= proc(n, k) option remember; `if`(n<0, 0, `if`(n=0, 1,
%p A216652       `if`(k<1, 0, b(n, k-1) +b(n-k, k))))
%p A216652     end:
%p A216652 T:= (n, k)-> b(n-k*(k+1)/2, k)*k!:
%p A216652 seq(seq(T(n, k), k=1..floor((sqrt(8*n+1)-1)/2)), n=1..24);  # _Alois P. Heinz_, Sep 12 2012
%t A216652 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
%Y A216652 Cf. A003056, A008289, A072574, A097910.
%K A216652 nonn,tabf
%O A216652 1,4
%A A216652 _Geoffrey Critzer_, Sep 12 2012