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%I #28 Apr 29 2020 07:44:57
%S 1,0,1,0,1,3,0,3,10,13,0,3,39,87,75,0,5,100,510,836,541,0,11,303,2272,
%T 7042,9025,4683,0,13,782,9999,46628,104255,109110,47293,0,19,2009,
%U 39369,284319,948725,1662273,1466003,545835,0,27,5388,154038,1577256,7676830,19798096,28538496,21713032,7087261
%N 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 colors in (weakly) increasing order; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%H Alois P. Heinz, <a href="/A327245/b327245.txt">Rows n = 0..140, flattened</a>
%F Sum_{k=1..n} k * T(n,k) = A327588(n).
%e T(3,1) = 3: 3aaa, 2aa1a, 1a2aa.
%e T(3,2) = 10: 3aab, 3abb, 2aa1b, 2ab1a, 2ab1b, 2bb1a, 1a2ab, 1a2bb, 1b2aa, 1b2ab.
%e T(3,3) = 13: 3abc, 2ab1c, 2ac1b, 2bc1a, 1a2bc, 1b2ac, 1c2ab, 1a1b1c, 1a1c1b, 1b1a1c, 1b1c1a, 1c1a1b, 1c1b1a.
%e Triangle T(n,k) begins:
%e 1;
%e 0, 1;
%e 0, 1, 3;
%e 0, 3, 10, 13;
%e 0, 3, 39, 87, 75;
%e 0, 5, 100, 510, 836, 541;
%e 0, 11, 303, 2272, 7042, 9025, 4683;
%e 0, 13, 782, 9999, 46628, 104255, 109110, 47293;
%e 0, 19, 2009, 39369, 284319, 948725, 1662273, 1466003, 545835;
%e ...
%p C:= binomial:
%p b:= proc(n, i, k, p) option remember; `if`(n=0, p!, `if`(i<1, 0, add(
%p b(n-i*j, min(n-i*j, i-1), k, p+j)*C(C(k+i-1, i), j), j=0..n/i)))
%p end:
%p T:= (n, k)-> add(b(n$2, i, 0)*(-1)^(k-i)*C(k, i), i=0..k):
%p seq(seq(T(n, k), k=0..n), n=0..10);
%t c = Binomial;
%t b[n_, i_, k_, p_] := b[n, i, k, p] = If[n == 0, p!, If[i < 1, 0, Sum[b[n - i*j, Min[n - i*j, i-1], k, p + j] c[c[k + i - 1, i], j], {j, 0, n/i}]]];
%t T[n_, k_] := Sum[b[n, n, i, 0] (-1)^(k - i) c[k, i], {i, 0, k}];
%t Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Apr 29 2020, after _Alois P. Heinz_ *)
%Y Columns k=0-2 give: A000007, A032020 (for n>0), A327847.
%Y Main diagonal gives A000670.
%Y Row sums give A321586.
%Y T(2n,n) gives A327589.
%Y Cf. A327244, A327588.
%K nonn,tabl
%O 0,6
%A _Alois P. Heinz_, Sep 14 2019
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