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A325915 Total number of colors used in all colored integer partitions of n where all colors from an initial interval of the color palette are used and parts differ by size or by color.

%I A325915 #25 Dec 09 2020 09:56:03
%S A325915 0,1,3,9,25,67,176,453,1149,2882,7161,17654,43238,105303,255210,
%T A325915 615896,1480771,3548313,8477415,20199596,48014369,113879450,269555798,
%U A325915 636875077,1502195104,3537705916,8319377813,19537936874,45827441193,107366261405,251268532266
%N A325915 Total number of colors used in all colored integer partitions of n where all colors from an initial interval of the color palette are used and parts differ by size or by color.
%H A325915 Alois P. Heinz, <a href="/A325915/b325915.txt">Table of n, a(n) for n = 0..2807</a>
%F A325915 a(n) = Sum_{k=1..n} k * A308680(n,k).
%F A325915 a(n) ~ c * d^n * n, where d = 2.26562663992642295791262530033324290454663... is the root of the equation QPochhammer[-1, 1/d] = 4 and c = 0.1771510533646387556482103930322780317974659818141571819... - _Vaclav Kotesovec_, Sep 18 2019
%p A325915 b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
%p A325915      `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
%p A325915     end:
%p A325915 g:= proc(n) option remember; `if`(n=0, [1, 0],
%p A325915       (p-> p+[0, p[1]])(add(b(j)*g(n-j), j=1..n)))
%p A325915     end:
%p A325915 a:= n-> g(n)[2]:
%p A325915 seq(a(n), n=0..32);
%t A325915 b[n_] := b[n] = If[n == 0, 1, Sum[b[n - j] Sum[If[OddQ[d], d, 0], {d, Divisors[j]}], {j, 1, n}]/n];
%t A325915 g[n_] := g[n] = If[n == 0, {1, 0}, Function[p, p + {0, p[[1]]}][Sum[b[j] g[n - j], {j, 1, n}]]];
%t A325915 a[n_] := g[n][[2]];
%t A325915 a /@ Range[0, 32] (* _Jean-François Alcover_, Dec 09 2020, after _Alois P. Heinz_ *)
%Y A325915 Cf. A000009, A304969, A308680.
%K A325915 nonn
%O A325915 0,3
%A A325915 _Alois P. Heinz_, Sep 08 2019