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A326617 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 distinct colors in increasing order; triangle T(n,k), k>=0, k<=n<=A024916(k), read by columns.

%I A326617 #50 Mar 12 2021 22:15:45
%S A326617 1,1,2,2,1,5,9,9,10,9,3,13,44,96,152,155,124,140,160,113,48,16,4,42,
%T A326617 225,680,1350,2180,3751,6050,7420,6870,5555,5330,6300,6475,5025,3000,
%U A326617 1250,250,150,1098,4155,11730,30300,69042,127364,188568,249690,365160,584733
%N A326617 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 distinct colors in increasing order; triangle T(n,k), k>=0, k<=n<=A024916(k), read by columns.
%C A326617 T(n,k) is defined for all n>=0 and k>=0.  The triangle displays only positive terms.  All other terms are zero.
%H A326617 Alois P. Heinz, <a href="/A326617/b326617.txt">Columns k = 0..40, flattened</a>
%H A326617 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%F A326617 Sum_{k=A185283(n)..n} k * T(n,k) = A326649(n).
%F A326617 Sum_{n=k..A024916(k)} n * T(n,k) = A326651(k).
%e A326617 T(3,2) = 2: 2a1b, 2b1a.
%e A326617 T(3,3) = 5: 3abc, 2ab1c, 2ac1b, 2bc1a, 111abc.
%e A326617 Triangle T(n,k) begins:
%e A326617   1;
%e A326617      1;
%e A326617         2;
%e A326617         2,  5;
%e A326617         1,  9,  13;
%e A326617             9,  44,   42;
%e A326617            10,  96,  225,   150;
%e A326617             9, 152,  680,  1098,    576;
%e A326617             3, 155, 1350,  4155,   5201,   2266;
%e A326617                124, 2180, 11730,  26642,  26904,   9966;
%e A326617                140, 3751, 30300, 106281, 182000, 149832, 47466;
%e A326617                ...
%p A326617 g:= proc(n) option remember; `if`(n=0, 0, numtheory[sigma](n)+g(n-1)) end:
%p A326617 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add((t->
%p A326617       b(n-t, min(n-t, i-1), k)*binomial(k, t))(i*j), j=0..n/i)))
%p A326617     end:
%p A326617 T:= (n, k)-> add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k):
%p A326617 seq(seq(T(n, k), n=k..g(k)), k=0..6);
%t A326617 g[n_] := g[n] = If[n == 0, 0, DivisorSigma[1, n] + g[n - 1]] ;
%t A326617 b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[With[{t = i j}, b[n - t, Min[n - t, i - 1], k]*Binomial[k, t]], {j, 0, n/i}]]];
%t A326617 T[n_, k_] := Sum[b[n, n, k - i]*(-1)^i*Binomial[k, i], {i, 0, k}];
%t A326617 Table[Table[T[n, k], {n, k, g[k]}], {k, 0, 6}] // Flatten (* _Jean-François Alcover_, Mar 12 2021, after _Alois P. Heinz_ *)
%Y A326617 Main diagonal gives A178682.
%Y A326617 Row sums give A326648.
%Y A326617 Column sums give A326650.
%Y A326617 Cf. A000203, A024916, A326616 (this triangle read by rows), A326649, A326651.
%K A326617 nonn,tabf
%O A326617 0,3
%A A326617 _Alois P. Heinz_, Sep 12 2019