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A326650 Number of colored integer partitions using all colors of an n-set such that each block of part i with multiplicity j has a pattern of i*j distinct colors in increasing order.

%I A326650 #25 Dec 09 2020 08:56:53
%S A326650 1,1,5,45,1065,61753,9705069,4394516773,5931440509137,
%T A326650 24154079629381105,300121111037478706517,11510717148660156841731485,
%U A326650 1369013994385630011763634779641,505666129597215709912984823873504809,582167751341290615329122568805084839847101
%N A326650 Number of colored integer partitions using all colors of an n-set such that each block of part i with multiplicity j has a pattern of i*j distinct colors in increasing order.
%H A326650 Alois P. Heinz, <a href="/A326650/b326650.txt">Table of n, a(n) for n = 0..50</a>
%p A326650 g:= proc(n) option remember; `if`(n=0, 0, numtheory[sigma](n)+g(n-1)) end:
%p A326650 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add((t->
%p A326650       b(n-t, min(n-t, i-1), k)*binomial(k, t))(i*j), j=0..n/i)))
%p A326650     end:
%p A326650 a:= k-> add(add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k), n=k..g(k)):
%p A326650 seq(a(n), n=0..15);
%t A326650 g[n_] := g[n] = If[n == 0, 0, DivisorSigma[1, n] + g[n - 1]];
%t A326650 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 A326650 a[k_] := Sum[b[n, n, k-i] (-1)^i Binomial[k, i], {n, k, g[k]}, {i, 0, k}];
%t A326650 a /@ Range[0, 15] (* _Jean-François Alcover_, Dec 09 2020, after _Alois P. Heinz_ *)
%Y A326650 Column sums of A326616 and of A326617.
%K A326650 nonn
%O A326650 0,3
%A A326650 _Alois P. Heinz_, Sep 12 2019