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A327584 Number T(n,k) of colored compositions of n using all colors of a k-set such that all parts have different color patterns and the patterns for parts i have i distinct colors in increasing order; triangle T(n,k), k>=0, k<=n<=k*2^(k-1), read by columns.

%I A327584 #25 Feb 22 2021 09:22:14
%S A327584 1,1,3,4,6,13,48,150,300,666,936,1824,2520,2160,5040,75,536,2820,
%T A327584 11144,41346,131304,420084,1191552,3427008,9207456,23466720,61522560,
%U A327584 141553560,345346560,777152160,1635096960,3700806480,6998261760,14211912960,27442437120
%N A327584 Number T(n,k) of colored compositions of n using all colors of a k-set such that all parts have different color patterns and the patterns for parts i have i distinct colors in increasing order; triangle T(n,k), k>=0, k<=n<=k*2^(k-1), read by columns.
%C A327584 T(n,k) is defined for all n>=0 and k>=0.  The triangle displays only positive terms.  All other terms are zero.
%H A327584 Alois P. Heinz, <a href="/A327584/b327584.txt">Columns k = 0..7, flattened</a>
%e A327584 T(3,2) = 4: 2ab1a, 2ab1b, 1a2ab, 1b2ab.
%e A327584 T(3,3) = 13: 3abc, 2ab1c, 2ac1b, 2bc1a, 1a2bc, 1b2ac, 1c2ab, 1a1b1c, 1a1c1b, 1b1a1c, 1b1c1a, 1c1a1b, 1c1b1a.
%e A327584 T(4,2) = 6: 2ab1a1b, 1a2ab1b, 1a1b2ab, 2ab1b1a, 1b2ab1a, 1b1a2ab.
%e A327584 Triangle T(n,k) begins:
%e A327584   1;
%e A327584      1;
%e A327584         3;
%e A327584         4,  13;
%e A327584         6,  48,    75;
%e A327584            150,   536,    541;
%e A327584            300,  2820,   6320,   4683;
%e A327584            666, 11144,  50150,  81012,   47293;
%e A327584            936, 41346, 308080, 903210, 1134952, 545835;
%e A327584            ...
%p A327584 C:= binomial:
%p A327584 g:= proc(n) option remember; n*2^(n-1) end:
%p A327584 h:= proc(n) option remember; local k; for k from
%p A327584       `if`(n=0, 0, h(n-1)) do if g(k)>=n then return k fi od
%p A327584     end:
%p A327584 b:= proc(n, i, k, p) option remember; `if`(n=0, p!,
%p A327584       `if`(i<1 or k<h(n), 0, add(b(n-i*j, min(n-i*j, i-1),
%p A327584         k, p+j)*C(C(k, i), j), j=0..n/i)))
%p A327584     end:
%p A327584 T:= (n, k)-> add(b(n$2, i, 0)*(-1)^(k-i)*C(k, i), i=0..k):
%p A327584 seq(seq(T(n, k), n=k..k*2^(k-1)), k=0..5);
%t A327584 c = Binomial;
%t A327584 g[n_] := g[n] = n*2^(n - 1);
%t A327584 h[n_] := h[n] = Module[{k}, For[k = If[n == 0, 0,
%t A327584      h[n - 1]], True, k++, If[g[k] >= n, Return[k]]]];
%t A327584 b[n_, i_, k_, p_] := b[n, i, k, p] = If[n == 0, p!,
%t A327584      If[i < 1 || k < h[n], 0, Sum[b[n - i*j, Min[n - i*j, i - 1],
%t A327584      k, p + j]*c[c[k, i], j], {j, 0, n/i}]]];
%t A327584 T[n_, k_] := Sum[b[n, n, i, 0]*(-1)^(k - i)*c[k, i], {i, 0, k}];
%t A327584 Table[Table[T[n, k], {n, k, k*2^(k - 1)}], {k, 0, 5}] // Flatten (* _Jean-François Alcover_, Feb 22 2021, after _Alois P. Heinz_ *)
%Y A327584 Main diagonal gives A000670.
%Y A327584 Row sums give A321587.
%Y A327584 Column sums give A327585.
%Y A327584 Cf. A001787, A326962, A327583 (this triangle read by rows).
%K A327584 nonn,tabf
%O A327584 0,3
%A A327584 _Alois P. Heinz_, Sep 17 2019