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A326500 Number T(n,k) of colored integer partitions of n using all colors of a k-set such that each block of part i with multiplicity j has a pattern of i*j colors in (weakly) increasing order; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%I A326500 #38 Apr 30 2020 11:36:23
%S A326500 1,0,1,0,2,2,0,3,8,5,0,5,22,30,13,0,7,54,129,124,42,0,11,118,428,696,
%T A326500 525,150,0,15,248,1293,3108,3830,2358,576,0,22,490,3483,11595,20720,
%U A326500 20535,10661,2266,0,30,950,9102,40592,99140,141234,117362,52824,9966
%N A326500 Number T(n,k) of colored integer partitions of n using all colors of a k-set such that each block of part i with multiplicity j has a pattern of i*j colors in (weakly) increasing order; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%H A326500 Alois P. Heinz, <a href="/A326500/b326500.txt">Rows n = 1..140, flattened</a>
%H A326500 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%F A326500 Sum_{k=1..n} k * T(n,k) = A326656(n).
%e A326500 T(3,1) = 3: 3aaa, 2aa1a, 111aaa.
%e A326500 T(3,2) = 8: 3aab, 3abb, 2aa1b, 2ab1b, 2ab1a, 2bb1a, 111aab, 111abb.
%e A326500 T(3,3) = 5: 3abc, 2ab1c, 2ac1b, 2bc1a, 111abc.
%e A326500 Triangle T(n,k) begins:
%e A326500   1;
%e A326500   0,  1;
%e A326500   0,  2,   2;
%e A326500   0,  3,   8,    5;
%e A326500   0,  5,  22,   30,    13;
%e A326500   0,  7,  54,  129,   124,    42;
%e A326500   0, 11, 118,  428,   696,   525,    150;
%e A326500   0, 15, 248, 1293,  3108,  3830,   2358,    576;
%e A326500   0, 22, 490, 3483, 11595, 20720,  20535,  10661,  2266;
%e A326500   0, 30, 950, 9102, 40592, 99140, 141234, 117362, 52824, 9966;
%e A326500   ...
%p A326500 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add((t->
%p A326500       b(n-t, min(n-t, i-1), k)*binomial(k+t-1, t))(i*j), j=0..n/i)))
%p A326500     end:
%p A326500 T:= (n, k)-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k):
%p A326500 seq(seq(T(n, k), k=0..n), n=0..12);
%t A326500 b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[Function[t, b[n-t, Min[n-t, i-1], k] Binomial[k + t - 1, t]][i j], {j, 0, n/i}]]];
%t A326500 T[n_, k_] := Sum[b[n, n, k - i] (-1)^i*Binomial[k, i], {i, 0, k}];
%t A326500 Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Apr 30 2020, after _Alois P. Heinz_ *)
%Y A326500 Columns k=0-1 give: A000007, A000041 (for n>0).
%Y A326500 Main diagonal gives A178682.
%Y A326500 Row sums give A326654.
%Y A326500 T(2n,n) gives A328158.
%Y A326500 Cf. A326656.
%K A326500 nonn,tabl
%O A326500 0,5
%A A326500 _Alois P. Heinz_, Sep 12 2019