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A327245 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.

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