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A309973 Number T(n,k) of colored integer partitions of n using all colors of a k-set such that parts i have distinct color patterns in arbitrary order and each pattern for a part i has i colors in (weakly) increasing order; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%I A309973 #39 May 29 2020 07:41:56
%S A309973 1,0,1,0,1,3,0,2,6,10,0,2,21,42,47,0,3,42,177,264,246,0,4,90,619,1746,
%T A309973 2095,1602,0,5,176,1809,7556,16085,16608,11481,0,6,348,5211,32621,
%U A309973 100030,171480,154385,95503,0,8,640,13961,120964,522890,1262832,1842659,1503232,871030
%N A309973 Number T(n,k) of colored integer partitions of n using all colors of a k-set such that parts i have distinct color patterns in arbitrary order and each pattern for a part i has i colors in (weakly) increasing order; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%H A309973 Alois P. Heinz, <a href="/A309973/b309973.txt">Rows n = 0..140, flattened</a>
%F A309973 Sum_{k=1..n} k * T(n,k) = A327680(n).
%e A309973 T(3,1) = 2: 3aaa, 2aa1a.
%e A309973 T(3,2) = 6: 3aab, 3abb, 2aa1b, 2ab1a, 2ab1b, 2bb1a.
%e A309973 T(3,3) = 10: 3abc, 2ab1c, 2ac1b, 2bc1a, 1a1b1c, 1a1c1b, 1b1a1c, 1b1c1a, 1c1a1b, 1c1b1a.
%e A309973 Triangle T(n,k) begins:
%e A309973   1;
%e A309973   0, 1;
%e A309973   0, 1,   3;
%e A309973   0, 2,   6,   10;
%e A309973   0, 2,  21,   42,    47;
%e A309973   0, 3,  42,  177,   264,    246;
%e A309973   0, 4,  90,  619,  1746,   2095,   1602;
%e A309973   0, 5, 176, 1809,  7556,  16085,  16608,  11481;
%e A309973   0, 6, 348, 5211, 32621, 100030, 171480, 154385, 95503;
%e A309973   ...
%p A309973 b:= proc(n, i, k) option remember; `if`(n=0, 1,
%p A309973       `if`(i<1, 0, add(b(n-i*j, min(n-i*j, i-1), k)*
%p A309973        binomial(binomial(k+i-1, i), j)*j!, j=0..n/i)))
%p A309973     end:
%p A309973 T:= (n, k)-> add(b(n$2, i)*(-1)^(k-i)*binomial(k, i), i=0..k):
%p A309973 seq(seq(T(n, k), k=0..n), n=0..12);
%t A309973 b[n_, i_, k_] := b[n, i, k] = If[n==0, 1, If[i<1, 0, Sum[b[n - i j, Min[n - i j, i-1], k] Binomial[Binomial[k+i-1, i], j] j!, {j, 0, n/i}]]];
%t A309973 T[n_, k_] := Sum[b[n, n, i] (-1)^(k - i) Binomial[k, i], {i, 0, k}];
%t A309973 Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, May 29 2020, after Maple *)
%Y A309973 Columns k=0-2 give: A000007, A000009 (for n>0), A327890.
%Y A309973 Main diagonal gives A005651.
%Y A309973 Row sums give A327679.
%Y A309973 T(2n,n) gives A327681.
%Y A309973 Cf. A327116, A327680.
%K A309973 nonn,tabl
%O A309973 0,6
%A A309973 _Alois P. Heinz_, Sep 21 2019