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A326649 Total number of colors in all 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 A326649 #24 Dec 09 2020 08:51:37
%S A326649 0,1,4,19,81,413,2439,14655,86844,573196,4224230,32280154,249433713,
%T A326649 1925416359,15732592327,139542345546,1304524118159,12445507282579,
%U A326649 119198874300879,1137647406084952,11183828252431175,116368970786569604,1278400213028604214
%N A326649 Total number of colors in all 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 A326649 Alois P. Heinz, <a href="/A326649/b326649.txt">Table of n, a(n) for n = 0..300</a>
%F A326649 a(n) = Sum_{k=A185283(n)..n} k * A326616(n,k) = Sum_{k=A185283(n)..n} k * A326617(n,k).
%p A326649 g:= proc(n) option remember; `if`(n=0, 0, numtheory[sigma](n)+g(n-1)) end:
%p A326649 h:= proc(n) option remember; local k; for k from
%p A326649       `if`(n=0, 0, h(n-1)) do if g(k)>=n then return k fi od
%p A326649     end:
%p A326649 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1 or k<h(n), 0, add(
%p A326649       (t-> b(n-t, min(n-t, i-1), k)*binomial(k, t))(i*j), j=0..n/i)))
%p A326649     end:
%p A326649 a:= n-> add(k*add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k), k=h(n)..n):
%p A326649 seq(a(n), n=0..25);
%t A326649 g[n_] := g[n] = If[n == 0, 0, DivisorSigma[1, n] + g[n - 1]];
%t A326649 h[n_] := h[n] = Module[{k}, For[k = If[n == 0, 0, h[n - 1]], True, k++, If[g[k] >= n, Return [k]]]];
%t A326649 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 A326649 a[n_] := Sum[k Sum[b[n, n, k-i] (-1)^i Binomial[k, i], {i, 0, k}], {k, h[n], n}];
%t A326649 a /@ Range[0, 25] (* _Jean-François Alcover_, Dec 09 2020, after _Alois P. Heinz_ *)
%Y A326649 Cf. A185283, A326616, A326617.
%K A326649 nonn
%O A326649 0,3
%A A326649 _Alois P. Heinz_, Sep 12 2019