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A240315 Triangular array read by rows: T(n,k) is the number of compositions of n into exactly k parts in which no part is unique (each part occurs at least twice).

%I A240315 #23 Jan 21 2023 02:23:26
%S A240315 1,0,0,0,0,1,0,0,0,1,0,0,1,0,1,0,0,0,0,0,1,0,0,1,1,6,0,1,0,0,0,0,0,10,
%T A240315 0,1,0,0,1,0,7,10,15,0,1,0,0,0,1,0,10,20,21,0,1,0,0,1,0,12,1,30,35,28,
%U A240315 0,1,0,0,0,0,0,20,0,56,56,36,0,1,0,0,1,1,13,10,126,21,98,84,45,0,1
%N A240315 Triangular array read by rows: T(n,k) is the number of compositions of n into exactly k parts in which no part is unique (each part occurs at least twice).
%C A240315 Row sums = A240085.
%D A240315 S. Heubach and T. Mansour, Combinatorics of Compositions and Words, Chapman and Hall, 2009 page 87.
%H A240315 Alois P. Heinz, <a href="/A240315/b240315.txt">Rows n = 0..140, flattened</a>
%F A240315 Product_{i>=1} exp(x^i*y) - x^i*y = Sum_{k>=0} A_k(x)*y^k/k!, where A_k(x) is the o.g.f. for the number of compositions of n into k parts in which no part is unique.  In other words, A_k(x) is the o.g.f. for column k.
%e A240315 Triangle begins:
%e A240315   1;
%e A240315   0, 0;
%e A240315   0, 0, 1;
%e A240315   0, 0, 0, 1;
%e A240315   0, 0, 1, 0,  1;
%e A240315   0, 0, 0, 0,  0,  1;
%e A240315   0, 0, 1, 1,  6,  0,   1;
%e A240315   0, 0, 0, 0,  0, 10,   0,  1;
%e A240315   0, 0, 1, 0,  7, 10,  15,  0,  1;
%e A240315   0, 0, 0, 1,  0, 10,  20, 21,  0,  1;
%e A240315   0, 0, 1, 0, 12,  1,  30, 35, 28,  0,  1;
%e A240315   0, 0, 0, 0,  0, 20,   0, 56, 56, 36,  0, 1;
%e A240315   0, 0, 1, 1, 13, 10, 126, 21, 98, 84, 45, 0, 1;
%e A240315   ...
%e A240315 T(8,4) = 7 because we have: 3+3+1+1, 3+1+3+1, 3+1+1+3, 1+3+3+1, 1+3+1+3, 1+1+3+3, 2+2+2+2.
%p A240315 b:= proc(n, i, t) option remember; `if`(n=0, t!, `if`(i<1, 0,
%p A240315       expand(b(n, i-1, t)+add(x^j*b(n-i*j, i-1, t+j)/j!, j=2..n/i))))
%p A240315     end:
%p A240315 T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2, 0)):
%p A240315 seq(T(n), n=0..14);  # _Alois P. Heinz_, Apr 03 2014
%t A240315 nn=10;Table[Take[Transpose[Range[0,nn]!CoefficientList[Series[ Product[Exp[x^i y]-x^i y,{i,1,nn}],{y,0,nn}],{y,x}]],nn+1][[j,Range[1,j]]],{j,1,nn}]//Grid
%K A240315 nonn,tabl
%O A240315 0,26
%A A240315 _Geoffrey Critzer_, Apr 03 2014