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A326648 Number of colored integer partitions of n using all colors of an initial interval of the color palette such that each block of part i with multiplicity j has a pattern of i*j distinct colors in increasing order.

%I A326648 #30 Dec 09 2020 07:55:12
%S A326648 1,1,2,7,23,95,481,2515,13130,77546,519770,3641724,25931163,185418629,
%T A326648 1411248697,11735504788,103340890753,931471895697,8448978391755,
%U A326648 76541843977198,715994685630321,7110500945450780,74757652968961770,815423663501064107,9012653697655462141
%N A326648 Number of colored integer partitions of n using all colors of an initial interval of the color palette such that each block of part i with multiplicity j has a pattern of i*j distinct colors in increasing order.
%H A326648 Alois P. Heinz, <a href="/A326648/b326648.txt">Table of n, a(n) for n = 0..300</a>
%p A326648 g:= proc(n) option remember; `if`(n=0, 0, numtheory[sigma](n)+g(n-1)) end:
%p A326648 h:= proc(n) option remember; local k; for k from
%p A326648       `if`(n=0, 0, h(n-1)) do if g(k)>=n then return k fi od
%p A326648     end:
%p A326648 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1 or k<h(n), 0, add(
%p A326648       (t-> b(n-t, min(n-t, i-1), k)*binomial(k, t))(i*j), j=0..n/i)))
%p A326648     end:
%p A326648 a:= n-> add(add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k), k=h(n)..n):
%p A326648 seq(a(n), n=0..25);
%t A326648 g[n_] := g[n] = If[n == 0, 0, DivisorSigma[1, n] + g[n-1]];
%t A326648 h[n_] := h[n] = Module[{k}, For[k = If[n == 0, 0, h[n-1]], True, k++, If[g[k] >= n, Return[k]]]];
%t A326648 b[n_, i_, k_] := b[n, i, k] = If[n==0, 1, If[i<1 || k < h[n], 0, Sum[With[ {t = i j}, b[n-t, Min[n-t, i-1], k] Binomial[k, t]], {j, 0, n/i}]]];
%t A326648 a[n_] := Sum[b[n, n, k-i] (-1)^i Binomial[k, i], {k, h[n], n}, {i, 0, k}];
%t A326648 a /@ Range[0, 25] (* _Jean-François Alcover_, Dec 09 2020, after _Alois P. Heinz_ *)
%Y A326648 Row sums of A326616 and of A326617.
%K A326648 nonn
%O A326648 0,3
%A A326648 _Alois P. Heinz_, Sep 12 2019